Applications of Generalized Nets

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ENERALIZED

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Editor: Krassimir T Atanassov Institute for Microsystems, Sofia, Bulgaria

World Scientific Singapore • New Jersey • London • Hong Kong

Published by World Scientific Publishing Co. Pte. Ltd. P O Box 128, Fairer Road, Singapore 9128 USA office: Suite IB, 1060 Main Street, River Edge, NJ 07661 UK office: 73 Lynton Mead, Totteridge, London N20 8DH

APPLICATIONS OF GENERALIZED NETS Copyright © 1993 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form orby any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

ISBN 981-02-0667-4

Printed in Singapore.

To Prof. Car]-Adam Petri on the occasion of the sary of

30-th anniver­

his beautiful idea na­

med Petri net.

PREFACE

This book contains papers on almost all models of processes de­ veloped

up to now,

Hets (GHs).

which are described on

the basis of

These extensions of Petri nets were

"Generalized Hets" (World Scientific Pub], Co.,

Generalized

described in my book Singapore, 1991). The

results used from the bock "Generalized Bets" are given in Appendix J, This book is divided in five areas: theoretical results on GHs, theoretical GH-models, GH-models of economical,

industrial and trans-

portational processes, GH-models of processes, related to medical dia­ gnostics,

and GH-models of computer processes.

Elements of the first

version of the Program Package for GHs (FPGH), which is already prepa­ red are described in the last chapter.

The authors are my colleagues,

postgraduate students and students, who are interested in GHs. Toe aim of this book is to show the largest possible scope of GHs. I hope that this task will be fulfilled.

applications of

GHs are realized in the

frames of FPGH. Those who are interested in them can receive information from me home address:

Bulgaria,

Sofia-1113,

detailed

ul. Elemag,

Ho. 15, Block 307, Ap. 18. ■ «

a

I would like to thank the Chief Editor of AHSE Review Prof G. Hesnard

(France)

for urging me to write this book and my coauthors for

the effort. Erassimir T. Atanassov April 1992, Sofia -

v

-

ThisThis page pageisisintentionally intentionally leftleft blankblank

CCSTEHTS

Chapter 1: Hew results in the theory of Generalized Hets (GBs)

1

§ 1.1. Hew results in the theory of GHs by E. Atanassov

2

§ 1.2. On the representation of GH transitions by a fixed set of transitions by P. Gyurov and X. Atanassov

31

§ 1.3. Two new conservative extensions of GHs by R. Cbristov and X. Atanassov Chapter 2: Some applications of GHs in science

43 48

§ 2. 1. GHs representing the elements of neuron networks by L. Hadjyisky and X. Atanassov

49

% 2.2. GHs representing the travelling salesman problem by K. Atanassov

68

§ 2. 3. GHs and labyrinths by F. Cbristov and X. Atanassov

82

§ 2.4. Forma] communication in science: a model based on GHs by R. Todorov and X

Atanassov

85

§ 2. 5. GHs and expert systems. V by X. Atanassov, F. Georgiev and H. Tetev

96

§ 2. 6. GHs researching the basic properties of Knowledge base by E. Dimitrov, I. Kazalarsky and X. Atanassov

106

§ 2.7. Parallel Knowledge processing modelling in expert system using GHs by I. Hristozov and E. Tzolov

112

§ 2. 8. GHs and distributed database technology by A. Georgiev

120

-

vii

-

Viii

CONTENTS

Chapter 3: Some applications of GNs in economics, industry and transport

124

§ 3. 1. Modelling the activities of international transport market by GNs by A. Kirova and K. Atanassov

125

§ 3.2. GNs for simulation of booking systems for passenger transport by S. Bedev and L. Atanassova

139

§ 3.3. GNs in the modelling of public transport by K. Stefanova

148

§ 3.4. GN model of activities of a bus-depot by P. Gyurov

154

§ 3. 5. GN model of the activity of an air-traffic control center by S. Fisbmanov

163

§3.6. GN models for flexible manufacturing systems by H. Stefanova-Favlova and K. Atanassov

172

§3.7. GN models of the activity of HEPTCCHIH Petrochemical Combine in Bourgas by S. Dimitrova, L. Dimitrova, T. Kolarova, F. Fetkov, K. Atanassov and B. Christov

208

§ 3. 8. A second type of intuitionistic fuzzy GN-model in the chemical industry by S. Christov and S. Garbov Chapter 4: Applications of GNs in medicine

214 219

§ 4.1. Applications of GNs in nepbrology by J. Sorsich and K. Atanassov

220

CONTENTS

ix

§ 4.2. Modelling of diagnostic and therapeutic processes in medicine by GHs by B. Jordanova and J. Sorsich Chapter 5: GHs and computer science

291 298

§ 5. 1: Implementation of GHs in distributed operating system development by H. Pehlivanov

299

§ 5.2: Gil-models of data transfer in industrial local area networks by A. Savov

306

§ 5.3: GH-modeIs of FFT and transputer systems by A. Dimitrova and R. Fetrov

314

§ 5.4: Implementation of GHs for simulation of node-to-node communication in a regular structure by S. Stefanov

319

§5.5: GN modelling of disk subsystems by B. Hikolova

327

§ 5.6: Software tools for GHs by R. Cbristov and S. Hihov

340

Appendix 1: Remarks on GHs (by Z. Atanassov]

349

Appendix 2: Generalized index matrices (by K. Atanassov)

379

Appendix 3: Intuitionistic fuzzy sets (by K. Atanassov)

382

List of authors

386

Chapter

1:

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

Hiree new results related to the theory of GHs this chapter.

They are continuations of

Ixiok "Generalized Bets".

1

are included in

the results from the

2

CHAPTER 1 § 1.1: NEW RESULTS IN THE THEORY OF GENERALIZED NETS Krassimir T. Atanassov

New results in

the theory of GNs,

mounts of 1991, are introduced below. from my booK

(noted here by [«];

obtained in

the first five

They are continuations of these

see Appendix 1).

A part of the re­

sults are based on [1, 2 ] . x

«

«

§ 1. 1. 1: HEmODGLCGICAL ASPECT OF 2Fffi 2HEDSF OF (ENERAL1ZED

SETS

Various mathematical tools are used for the development and ex­ tension of the theory of GNs.

Algebraic,

her aspects of the theory of GNs were

logical, topological and ot­

introduced during

the last ten

years. GNs are used as tools for describing some mathematical problems and some real processes. GN's application is possible in the following areas: (a) tools for modelling,

simulation and forecast of the processes,

(b) real-time control and/or optimization of non-critical

in time

real-time processes. Figure 1. 1 represents the general scheme of the GN's applicati­ ons.

when the real process modelled by a GN is complex enough,

it is

possible to use a combination of two or all the three functions of the GN-model. After the construction of a given GN-model, questions about its validity and adequacy are important. In this case, it is necessary to make a comparison with the re­ ality.

To this aim

we need some validity criteria.

with GNs must define such criteria, for

their results.

Users

modelling

which give them "enough evidence"

There exist different

methods for

aim. An idea for such a criterion may be as follows:

reaching this

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

3

Fig. i. 1 Let the sets K , K 1 2

K

c K(E) for s € N, be randomly chos

sen; let, for every i (l s i <_, s), the sets X

a z {x / a € K ! c X(K ) i 0 1 1

be randomly chosen also, and let Y

a = (x / a e K 1, 1 fin 1

o a where x and x are the initial and final characteristics of the to0 fin s ken a, respectively. Let the function F: (P(K(E))) -> P(K(E)> be such that F(K , K 1 2

s K | : K c U K C K(E), s 1=1 1

4

CHAPTER 1

where P(A) is the set of all subsets of the set A.

The above property

of the function F can be considered in a way as an

"averaging proper­

ty". Toe sets a X = |x (x // aa f€lK] l 0 and a Y : = (x / a € K] fin can be constructed directly and let a a Y = (x / (3i: 1 i i i i s) s] (a € K » x 6 €"»}!. Y } J. fin i fin i Let P(Y, Y) be a criterion for "closeness"

between the sets

Y

and Y, which is defined by (i, if the sets Y and Y are "close" (in some sense), P(Y, Y) -- \ ( . 0, otherwise. If for a number

s

large enough, and the equality P(Y, Y) -- 1

is valid, we say that GH

E is statistically significant (in the other

case - it is non-statistically significant). If a given GH E is stati­ stically

significant, we can assert that every result of its functio­

ning over a set of tokens stics

X' c X(E)

KJ c K(E) with a set of initial characteri­

is statistically significant.

There are other ways of constructing this criterion. In GH-modelling, the form

it is possible

that the adaptive method with

from Fig. 1.2 is the most positive one.

In this case,

the

corrections in the constructed GH-model are done adaptively on the ba­ sis of the difference

between the

expected and obtained results

relation with the same fixed criteria),

(in

we must bear in mind that the

constructed GH-model is realized by the same program (e.g. in the fra-

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

mework

of a "Program Package for GNs"

PPGN,

now being

5

developed as

shown in Fig. 1. 2; any other program tool which realises GNs the place of "PPGN"). rections

Some errors can occur in this

may be necessary,

present description.

but this

is not in

can take

program and cor­

the framework of the

Thus, we do not discuss the possibility

rections in the program tools for GN realizations. However, ists a probability to realise this too,

if the results

of cor­ there ex­

show existing

defects in the programs. As shown in Fig. 1. 2,

the process is simulated many

times and

the results are conpared (and eventually averaged), analysed and esti­ mated. An analogy of

the above scheme

can be used

when research

is

carried out with Petri nets or their extensions. But, for example, al­ gebraic operations over GNs

and hierarchical operators over

GNs (see

§ 5 and § ft in App. 1) give possibility of more ccunplex models.

These

GN-models can be constructed by two main types: - union, - hierarchical. To describe different

processes which function in parallel, we

can construct GN-models for each one of them and thus construct the GN which is a union of all these GN-models. After this we proceed as in the above (sinple) case.

Now, the correc­

tions of the model are made in the general GN (see Fig. 1.3). There are two

basic ways of constructing the general GN in the

frames of the union types: - simultaneous, - sequential. In the case of the simultaneous union type, the different (ele­ mentary) GNs are constructed preliminarily and the new (general) GN is obtained as a union of them The validity of

the new GN is checked as

6

CHAPTER 1

shown in

Pig. 1.2 and the corrections are made in the genera] GH (see

Fig. 1.3).

Fig. 1.2 In the case of the sequential union type, we give priorities to the different (elementary) GHs, and we start constructing the new (ge­ neral)

GH as a union of the current particular general GH and

the GH

with a current number. Initially, the particular general GH is an emp­ ty GH and the GH with a current number is the GH with the highest pri­ ority; finally,

the particular general GH is

the same general GH and

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

7

the GN with current number is the GN with the lovest priority). In every step of the last process of GN-constructing, we make a checK for the validity of the functioning of the particular general GN and, if necessary, make corrections in the GN with the current number, because

the other GN

has already been tested.

Thus,

the number

of

tests is larger, but the corrections are made in simpler nets. The other extension of the scheme from Fig. 1. 2 (when the GN is obtained by hierarchical type ways) is shown in Fig. 1.4.

In this ca­

se, there exist the same two types: - simultaneous, - sequential, and

the constructions

rence between

are similar to the above ones. The main diffe­

the hierarchical type sequential way and the union type

sequential way is as follows: current particular general

GN

in the second case,

vel, while in the first case the growth is in of the levels and growing of the subnets on ces or

the growing of the

is on the basis of nets with equal le­ two directions: growing

the same level (i.e. pla­

transitions of the basic GN are changed

with new subnets,

in

which it is made the same, etc. : places or transitions of a subnet of the GN from the last level are changed with new subnets). Note, that there is no a restriction on rent particular general ally,

the growth of the cur­

GN on levels and in the level, i. e. sequenti­

different new subnets on

the same level,

as well as different

subnets from different levels, can be added to it. Nets, which describe some processes liKe the process of the so­ lution of a transportation problem (see §3.7 [»]), the process of the solution of a travelling salesman problem

(see § 2.2)

and others can

be included in the frames of the addition to a fixed GN. These GNs are tested preliminarily as independent

GNs and can be added to different

GN-models. In future, we shall construct GNs which describe some sche-

8

CHAPTER 1

Fig. 1. 3

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

Ailing problems, other modifications of

9

the two mathematical problems

mentioned above and other optimization problems. The above

schemes illustrate one of

the ways of using

GN-mo-

dels: GN-simulation of the modelled processes. The confirmation of the validity can be obtained after more si­ mulations of the process with the GN-model and after comparing the re­ sults.

Thus we can say that the constructed GN-model is statistically

reliable. With these

remarks, we shall briefly

describe the process

of

J

GN-model s construction. Then, the GN-model can be used, as a tool for simulation, control and optimization (of the modelled real processes). We shall show some approaches of using GNs for these aims, The first of these approaches is the sequential. the basis of the GN-model,

Initially, on

the corresponding real process can be con­

trolled, e. g. with the help of an expert or a dispatcher. This is pos­ sible (see § 8 in App. 1) by virtue of the check on the suitable token transfer strategy. Different data related to the modelled process are accumulated memory)

(as token's characteristics and as data in the global GN-

in the GN-model

in the framework of the process control. The

accumulated information (from the necessary number of seances) is ana­ lysed, estimated and manipulated by statistical means and on the basis of these, the values of some parameters of the random functions ted to different

transition condition

different characteristic functions ces)

are determined.

(with the same will be reached

predicates and

graphical structure)

the values

(which correspond to some

On this basis we can

rela­ of

GN-pla-

construct a new GN-model

of the process

(the alternation

only with the help of a suitable dynamical

operator)

which can simulate the modelled process. A variant of this procedure is as follows. Initially, the valu­ es of the parameters in the

random functions

which are in the places

10

Fig.

CHAPTER 1

1.4

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

of some transition condition predicates The results of the simulation process these values are changed with

others.

are determined

11

by an expert.

are compared and, if necessary, After this, the control of the

real process starts and the above procedure is repeated In both cases the process of precising the random values of the transition condition predicates and the place's characteristic functi­ ons can be an adaptive one (as in Figs. 1. 2 and 1. 3). This situation,

although more complex,

will be valid when the

GN-model is also used as an optimization tool. To illustrate the above reasonings

we shall discuss the possi­

bility of using GNs as a tool for describing of global processes. The­ re are two approaches: ascending and downgrading. The ascending approach is based on

the above discussed

proce­

dures for the GN-model constructions. For example, if we have to model the functioning of some chemical works, we can initially construct the GN-models

of the different work sections,

and later construct the GN

which corresponds to the work in general.

In this net

the token will

receive

as current characteristics those

values which

the tokens of

the GNs

corresponding to the different work sections

the final characteristics;

will receive as

and each of these sections will

be repre­

sented in the global GN by a certain place or a certain transition. The downgrading approach is the opposite. Initially, ruct the GN,

we const­

which represents the basic sections of the work

and the

relation between them (the general GN) and later we start to construct the GNs which correspond to the different works' sections. In this ca­ se we mist use the expert information about

the works' processes when

we determine the transition condition predicates and the place charac­ teristic functions. In a way, the places

of the general GN are "black

boxes". Finally, the formulation

we shall discuss

the possibility to use

of mathematical problems,

the GNs for

and more precisely

(here)

12

CHAPTER 1

for the generalization of such problems. We shall use see

the first GH from § 2. 2 as the first example.

that in this GN we do not use all the GH's

delling. Actually, the token which represents the travelling can get some characteristics person. pes ness. can

He

We

possibilities for mo­

related to the travelling

salesman

salesman as a

(the man, represented as a token) can carry different ty­

of products in corresponding quantities, The built-up areas where the be specified

prices,

limits of fit­

travelling salesman can also come

by the nesessary quantities of

products for

them,

distances between places and costs of transport between them, if there exists a relation between them

and others.

The number

of travelling

salesmen can be more than one (as in some generalizations of this pro­ blem) but now we can discuss the problems in which the travelling

sa­

lesmen are representatives of one or more firms, i.e., they are or are not in competition among themselves.

In § 2. 2, a GH which describes a

travelling salesman problem in some of these new modifications is dis­ cussed (the second GN). Our second example will be related to the concept tem", which

is a basic element of the new and modern scientific

"artificial intelligence".

In a series of papers

GHs which describe the functioning and expert system from a given class

results

Three

"data base"

enter the most complex and general

area

(3-6] we construct

of the

expert systems.

initial characteristics: "hypothesis", les"

"expert sys­

work of every tokens

and "list

one of these GHs,

with of ru­

which are

described in [6]. The first token is split during the process of a GHtransfer into several different tokens and each of them leaves ttoe net with a characteristic "the hypothesis is valid" or not valid".

"the hypothesis is

In the first case, the second token receives as

characteristic the initial

the next

characteristic of the first taken and

process continues (at a given moment

the

other tokens with other hypothe-

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

13

ses can enter the net). The third token is not modified during the ti­ me

of GN-functioning.

This is the real situation in

the expert sys­

tems too. On the other hand, there exists some research related to the addition of new rules

to the Knowledge base and this process can also

be described by a certain <3N. Thus we can construct a new GN which de­ scribes both processes when they function in parallel. This is not re­ alized in the expert systems yet. Moreover, the token of the third type from the new GN can recei­ ve

a current characteristic in relation to

of the first two types.

the values of

the tokens

Thus, on the basis of GN-models, self-modifi­

cation expert systems can be constructed. For example, time-components can also be

added to these expert systems.

ject of another research.

This idea will be the ob­

The generalization of the neural

networks,

introduced in [7] on the basis of [8] (cf. § 2. 1) is the third example of the application of GNs as a tool

for the generation of

of existing processes, problems and concepts. sion of

extensions

The idea of this exten­

the neural networks (in the frames of a given neural network,

new neurons are generated with connections between them and some other neurons)

is introduced after comparing the GN-components

used in the

GN from [8] to the other un-used GN-possibilities. we can produce in an analogous way some other GN-models too. GN-interpretations

of some ideas from

[9-15]

and other books

will be discussed in a next methodological paper on GNs. * »

«

§ 1. 1. 2: MORE Off US! CONCEPT "GENERALIZED J\E7S"'

One more definition of the concept GN will be introduced here. All other definitions of this concept are very complex. the concept reduced GN, E, having the form

we shall discuss

Now, based on

the properties of an object

14

CHAPTER 1

E - , where (a) A is a set of transitions, each one of them having the form Z = , where L' and L" are the sets of the input and r

output transition places,

is the transition condition;

(b) K is the set of tokens; (c) X is

the set of token's initial characteristics,

from App. 1, we can also delete the component X,

%2

(following

but in this case

the description will be complicated), (d) $ is the token's characteristic function. All GHs J elements stated above are as within the definition

of

ordinary GHs. Hote that this definition, similar to the ordinary OH definiti­ on, is not fully formalized. If we fully formalize the transition con­ ditions

and characteristic functions of the GHs,

a smaller class

of

GHs will be obtained. Obviously, a GN defined in such a way is reduced dinary elements of the class £

about the or­

of all GHs. On the other band (cf. § 2

in App. 1), each ordinary GH can be represented by a certain

GH with

the above form. Therefore the above GH has the modelling possibilities of the other types of GHs. There are processes which by

can be described

simpler types of GHs and in some cases, the reduced

GHs described

above are suitable as tools for modelling. Below

we shall

discuss the basic properties

GHs, which we can call GHs, without restriction. Let of all such GHs.

of

these reduced

£ J be the

From this point of view the ordinary GHs

class

can be ac­

cepted as conservative extensions of the reduced one and £ i- %*, The algorithms for token's transfers

in the frames of

a fixed

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

transition

(for

the abstract type

functioning of a fixed ordinary GN

15

of transitions also) can also be

and for the

used for

the reduced

GN (without essential corrections). By analogy with ordinary GNs, we can construct fuzzy itionistic

fuzzy (reduced) GNs from both types, colour GNs,

interval time for activation,

GNs with a complex structure,

and intuGNs with GNs with

an optimization component, GNs with a global memory, GNs with an addi­ tional clocK. It is important to note, that all other Petri net modifications are represented by the

(reduced) GNs,

having almost the same form as

the ordinary GN. We shall discuss

in detail the operations

£'. We shall repeat the definitions from

and relations

over

§ 5 on App. 1 with the cor­

responding changes, For the two transitions Z

and Z 1

of a fixed GN E, we shall de2

fine Z 1 Z 1

c Z 2

= Z 2

iff (Vi: 1 i i i 3)(pr Z = pr Z ); i 1 i 2

iff (Vi: 1 i i i 2)(pr Z c pr Z ) & i 1 1 2

(pr Z c 3 1

pr Z ), 3 2

where c

is a relation of inclusion over index matrices and if

A = [K , L , 1 1

la |], B = IK , L , lb |], then i, j 2 2 i, J AC

B iff (K c K ) & (L C L ) 8, (Vi€K )(VJ€L )(a -b ). 1 2 1 2 1 1 i, J i, J

Over the transitions Z i three operations.

- i i i

It is necessary

(i = 1, 2), we shall define

for the following to be valid:

place 1 € pr Z D pr Z and 1 € pr Z , then i i 2 i s 3-i i 2, 1 i s i 2. These operations are

1 € pr

if

Z for 1 i i 3-s 3-i

16

CHAPTER 1

(a) a union

(the necessary condition lor

this operation is: if place

1 6 pr Z , then it is not possible, that 1 € pr Z for 1 1 i i s i 3-s 3-i 2, 1 i s i 2) Z Z

1 2 2 1 22 11 22 = =<; >; 1 1 11 2 2 22

U ZZ 1 2

(b) an i n t e r s e c t i o n (with the above c o n d i t i o n s ) 1

Z flZ 1 2

=
2 1 2 1 2 n L , L ( 1 L , r xx r >>;; 1 1 2 2

(c) a composition (with the above c o n d i t i o n and w i t h the c o n d i t i o n 1 L 1 Z oZ 1 2

=
2 (1L 1 1

1 2 = L n L = (0, 2 2

2 1 2 1 2 U ( L - L ) , L U(L - L ) , r 1 2 2 2 1

As i n § 5 i n App. 1,

we s h a l l f o m u l a t e seme

1 2 o r > . c o n d i t i o n s which

t h e arguments of t h e s e o p e r a t i o n s ( d i f f e r e n t GHs) most s a t i s f y . We s h a l l assume t h a t two n e t s ,

i f a place takes

part simultaneously

in

then i n both l o c a t i o n s the p l a c e has

(a) the same characteristic functions, (b) equal possibility of accepting tokens from X, i.e. if in the first net (which is a model of a certain process) the tokens, elements of the set K' c K can pass

which are

through the examined place,

and in the second net the tokens which are elements of the set K" c E can pass through this place, then K' = K", (c) one and the same numeration/notation. (d) equal data in these tokens of both nets, ons about

both places

which contain informati­

(i.e. equal values of the place's priority

function and of the place's capacity function). Similarly,

we shall expect that if two places are respectively

an input one and an output one for two transitions in both nets, the tokens of both nets, which contain informations about

then

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

17

(a) the capacities of the connecting arcs, (b) the time for passing from the input to the output place, must have equal data.

Also, these places should be in the same consequence,

i.e. the place which is an input (output) one in the first of the transitions should be the same in the other transition too. Let E

and E 1

be two GNs and for i = 1, 2 let: 2 E - . >. ii ii ii ii ii

A union will be called the object E 1

UE 2

=
UA,K UK.X U X , $ 2 1 2 1 2

U $ >, 1 2

where 2 Oo A U A = U (Z/(Z € A ) & (V Z' € A )(Z D Z' - Z )} U 1 2 i=i i 3-i 2 o U IZ/(3Z' 6 A )(3Z" € A ) (Z' n Z" i. Z ) & (Z = Z' U Z")IZ")I. i=l 1=1 i 3-i Here the operations "union" and "composition", defined over or­ dinary GNs, coincide with the above GNs. Local operator 0 and operation "iteration" in § 5 in App

l are

defined in the same way. For each two GNs E 1 E ; : E 1 22

iff i f f (Vi (Vi 1 1 i i i 5)<pr 5)<pr E = pr E ), ), ii 1 1i 2

E C E 1 x 2

iff

(VZ € A )(3Z 1 1 2 & (X 1

where

and E , having the above forms, 2

g : hIE

£ A )(Z C Z ) & (K 2 1 2

CK ) 1 2

C X > & ($ = $ IE ), 2 1 2 1

denotes that the function

g

is a restriction of

the

1 function h over the first GN E . Let for both functions 1 Dom(g) - Dom(h):

g and h

with

16

CHAPTER 1

g i h Iff E

= E iff o 2

1

(Vx € Dom(g)) (g (x) = h ( x ) ) .

(Vi 1 i i i4 3 ) ( p r E = pr E ); i 1i l i i 2 I

E Jc

E

iff

(A c A j 1 2

1 IMQ

c' E i f f 1 o 2

(A c A ] 1 2

t IMQ

o

2

I c QQ ) )« 8( K(X 1 2

O E

the relation E

c 1

E

c 1

E

E o

c 1

U

J

iff (E <E 2

1

E iff 2

E o

O C Q ) 8S (K (X 1 2

c K ), 1 2 CK], 1 2

is not defined here, 2

c o

E ) v (E c' E ). 2 1 o 2

(A c A ] 8 (K c X ) 8 (X C X ) 8 <$ 1 2 1 2 1 2

For a given token

a €X

= i IE ] , 1 2 1

and a given initial characteristic

x € X

let

f

a

, if a € E and x € X fin E(a, X) = <

, otherwise

a /■ aa a a a where x fin

>, if a € E and x € X ,

, otherwise

is the final characteristic of the token a in the GH E.

Let us define E (a, x) - E |a, x] = . Below we shall introduce four other relations between GHs: E 1

c c E 22

iff i f f (X c c X X ] ] 8 (X cc XX )) S&( (Va V a €€ XX )) ((Vx V x€€ XX ) ) 1 22 1 22 1 1 (E (a, (E (a, x) x) = E (a, (a, xx)), )), 1 2

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

E cC EE iff 1I « 2

(K 1

C K ) & (X C X I MVQ £ K | 2 1 2 1

(Vx € X ><3i , i 1 1 2

i : 1l si i < .. . .. << i i i 1 s s

(E (a, x! = pr 1 i ,i 1 2 E 1 E

8 E 2

iI f f

(E 1

« E iff 1i * » 2 2

(E

i9

i

fin)

E (a, x) x)>, >, 2 s

c E ) & (E c E ), 2 2 1 c E ) & (E c E ). 11 ** 22 22 x* 11

Obviously, the relations c «

and » *

are stronger than the rela-

tions c and «, respectively. The examples and assertions from Chapter 5 [*]

are valid with­

out essential corrections, Dut the proofs are (obviously) simpler. The theorem for the completeness of the GN's transitions is also valid The complexity

operators

applied to the

(reduced)

3

GN without change.

5

7

This is valid

are

13 for

all

graph-operators too. All assertions from Chapter 6 [*] are valid here without essen­ tial corrections (cf. § 6 in App. 1). The same is valid for non-temporal logical operators from Chap­ ter 7 [»]. The problem with the application

of different types

of opera­

tors is more complex. Some of the global ones, G

, and G II

rectly",

can be applied directly.

as G , G , G , G , G , 1 2 3 4 6 Others are applied only "indi-

13 i.e.,

in the following way: when a corresponding (ordinary)

GN's component (which is represented in the (reduced) GN with the cha­ racteristic

of a certain

special token) must be

changed by some of

these operators, this can be done by operator G . Local operators P , 12 5

20

CHAPTER 1

F ,F , P 8 9 10

are applied

operator G

directly and the others

are

represented by

The hierarchical operators are modified for the present 12

case by the means discussed above. The reduced operator is defined only over the set f, X, X, $] A I

and, as

in

the case

with the

(A , A , A , 1 2 5

ordinary GHs,

the

sets

A 1 2 A X . I . I . Z

are empty.

All extending operators (with necessary additions) can be defi­ ned over the (reduced) GH too.

Horeover, we can now

ding operator, which will juxtapose a new

define an exten­

ordinary GH to every (redu­

ced) GH. As shown above, this extension is conservative. when the information

about arc's capacities

some token's initial characteristics,

al1 A

tors (1 s j £ 18) can be applied over the band, the operators A

is determined

-type dynamical 1. J

(reduced) GH.

are applied only where

in

opera-

On the other

j 6 12,5,8], The ope-

1, j rators of A

-type where 2 i i i 1 and

j

is arbirtarily admissible,

ii J

are applied directly. Based the above definitions we can construct self-modifying re­ duced GHs and universal reduced GHs. Finally, we shall emphassize that GHs are simpler than ordinary ones,

but more processes exist whose models by

more convenient.

• *

«

ordinary GHs are much

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

§ 1. i. 3:

UNIVERSAL GENERALIZED JSE7S.

21

II.

We shall discuss two problems related

to the concepts of Uni­

versal GNs (UGNs> and Self-Modifying GNs (SM3Ns)

(see § 9 in

App. 1)

and to the GN-problems from § 11 in App. 1: 1. the existence of a Universal GN (UGN) for the class 2. the existence of a UGN for the class £

£

of all GNs,

of all SMGNs.

SMGN In

§ 9. 1 [x] we have shown that if

a fixed

UGN

had started

functioning before a given GN, then the UGN may not to be able present the functioning of rem 1 [17]

the given GN. However,

to re­

the proof of Theo­

is wrong. Following Theorem 2 [17] and Theorem 1 of § 9 in

App. l, we construct a UGN for which (a) T = maxfT / (3E |3E € £ £)(T r = pr pr E)i 1 3 (it is possible: T = - oo); -o o -o (b) t - minft / (3E e £|(t = pr pr E)l E)] 2 3 -o (everywhere t > 0) and -K

(c) t

K

-K

= maxft / (3E € £)
= pr pr E)] 3 3

-K

(it is possible: t then for each GN,

= + co);

the GN from Theorem 2 [17] and Theorem 1 of

App. 1 will be a UGN, i.e. (-co)-UGN of the class

§ 9 in

(this is the author's mistake in [17]) the

£

is a UGN for the class

£.

This GN will

-oo function at every moment

and its

tokens will be as the tokens of the

UGN from Theorem 2 [17]. Therefore the following is valid: THEOREM 1. 1. 3. 1: There exists a GN which is a UGN for the class £.

22

CHAPTER 1

All other r e s u l t s from [17] are v a l i d . Hence f o r every s u b c l a s s of GHs, a UGH e x i s t s

which r e p r e s e n t s the f u n c t i o n i n g and

r e s u l t s of

the work of each element of t h i s s u b c l a s s . THEOREM 1, 1, 3. 2: There e x i s t s a GH, which i s an UQH f o r £ SHGH

« Proof: Let the GH E

from Fig.

1.5 be g i v e n and 1 4

r

1 5

= 1 1 W 1 1 1 1 1 | 4 1

1 16

false

W 3

W 4

W 2 false

where

,

W = "the token i s a" 1 (We assume t h a t card(K) > 0 because i n the o p p o s i t e case, t h e GH E i s not v a l i d . ) | W

= nW ;

22 11 W = "c(l , TIME) > 1"; 3 1 W w

= = nW iw :;

4

3

D D

= M il ■ i1 ); 11 44 1 6

11

r

= 1 2

I 1

5 1

I 11 I 11 I

W 5 W 5 5

where

p

r

W = " ( 3 Z € p r p r x P 1 (pr Z = ir ), W5 = " ( 3 Z € p r 1p r 1x cu 1 (pr3 Z = i cu ), 5 1 1 cu 3 cu 0D =- v ( l , 1 ); 2 5 11

NEW RESULTS IN THE THEORY OF GENERALIZED NETS 1 7 1 I 6 I r 3

= 1 2 1

1 8

true

1 9

false

false

I false 1

W

W

I false

W

6

7 w

12 I 6 7 where where W = "in the last characteristic of the token from 1 there is at W 6 = "in the last characteristic of the token from 1e there is at 6

6

least one GN-control token"

least one GN-control token" (This is possible because the token a has the highest priority. ); (This is possible because the token a has the highest priority. ); 7 6

W

: 1W ;

7

6

D 1 1 I 7 i

10

W 7

3 1

= M l , v(l , 1 )); 6 2 12 1 1 1 11 12 13

15

W a

false

false

false

false

W

W

false

false

false

false ,

w

w 8

1 8

i false 1

false

r = 1 4 9

I false 1

false

W

1 3

| false 1

false

false

1

|

false

false

1 14

8 8

7 W

7

false

7 false

15 I

false

W 7

\t*iere r 9 0 W - "x < pr pr x + pr pr x ", 8 cu 1 3 0 3 3 0

w = -m , 7

8 D -. A ( 1 , v ( l , 1 ), v ( l , 1 >); 4 7 8 9 3 15

w 8

23

24

CHAPTER 1

r 5 5

= 1 I 10 I 10 I

1 17 17

1

W

W 9 9

1 I false 16 I

18 18 10 10 true

where W

= "c(l 9

W 10

, TIME) > 0" 16

= nW , 9 D

= Mv(l 5

, 1 ), 1 ). 10 17 16

Let an arbitrary SHGN E with a set of tokens K = |a, a 1 2

K he given, where

a ). I]

« Let the tokens of E enter place 1 of E 1

(with the

E-charac-

teristics, i.e. with their initial E-characteristics). Let the token a with the ira-gimm

possible priority

(e.g.

n (a) --co) be also in 1 K

without initial characteristic. tic

"the description of

1

Token p with an initial characteris­

E"

enters place 1 and the token r with an 2 initial characteristic pr pr E (= T - the initial time of the functi1 3 oning of E) enters place 1 . All E-tokens from 1 enter sequentially 3 1 in place 1 without characteristics. Token a enters place 1 and the16 4 re it receives sequentially as

a characteristic: "the characteristics

of all E-tokens and their priorities" (in their priority-order). After that, the token a enters place 1 without a characteristic. The E-to5 * kens wait for the end of the process of functioning of E in place 1 16

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

Fig. 1. 5

25

26

CHAPTER 1

There they will receive their final characteristics. The priorities of the three tokens (a, p and r) satisfy the condition n (a) > n n
a

enters place

1

with a

list of

these E-tokens

6 which are transferred in the current

time-moment from input to output

E-transition places and a list of the corresponding characteristics of these E-tokens. For a more detailed description of this process, the place 1 mast be changed (by the hierarchical operator H with a GH 6 1 from the type of the GH from Theorems 9.1.1 of § 9. 1 in [«]. We do not « describe precisely this element of the net

E

because it

will

only

complicate its description. As a result of the activation of the transition Z , 3 a and p enter places

1 and 1 or 1 , respectively. 7 8 9

the tokens

In place 1 7

the

token a will not receive a characteristic. The token p will enter pla­ ce 1 , if in the current characteristic of a there exists at least one

a special E-token; otherwise p enters place 1 . In

the first case p re-

9 ceives as

a new

characteristic the

new description of

the SHGB E,

whose description is received as a result of applying operator(s) over E

(the type of this (these) operator(s) is (are) determined as a cha­

racteristic of the token a. In the second case, new characteristic. Transition Z 4

p

is activated when the three

does not receive a tokens are in

its

input places. After this, token a enters place 1 or 1 without cha10 11 racteristics, token p enters place 1 or 1 without characteristics, 12 13

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

27

token r enters place 1 or 1 without characteristic in 1 and with 14 15 14 a characteristic "the new value of the current time" in 1 , in rela15 tioin to the value of the current time. Finally, transition Z

is activated when the time for the func5

tioning of the SMGN E expires. The token a

(which has a higher pri­

ority over the tokens from place 1 ) enters place 1 without a cha16 17 racteristic. The tokens from 1 enter sequentially place 1 and the16 re

they receive

1A

as characteristic the

list of

all

characteristics

which are related to them The list is received as a characteristic of K

the token a at the time of its transfer in the net E . Thus the E-toK

kens a ,..., a will leave the G N E with 1 n

the same

characteristics

as those tokens leaving the SMGN E. Therefore the new net will represent

the functioning

of

the

SMGN E. We shall denote for E

as follows: all its places and transiti­

ons have equal priorities; the places

1 , 1 1

capacity; the others, capacities 1. The GN time-components can be (a) T ; 0; (a) T ; 0; o
and 16

1

have infinite 18

26

CHAPTER 1

where a € la, p, rii
a way represents

the functioning

of

each SHQB and therefore it is a UGH for £ SHGH Therefore Theorem 1. 1. 3. 2 rem 1. 1, 3. 1 is also given. THEOREM i. 1. 3. 3: £ H L SHGH

i.e.

is proved and another proof of Theo­ £ SHGH

i s a conservative

extension

of £. Proof: From Theorem 1. 1. 3. 2,

it follows that each SHGH

sented by the UGH, which is an element of of £

can be represented

£.

by an element of

can be repre­

Therefore every element £ and hence

£ I- £

SHGH

SHGH

The other direction is obvious. Thus we give an answer of the Problem 19 in App, 1. REFEREHCES: [1] E. Atanassov

Methodological aspect

of the theory of generalized

nets. II, AHSE Review Vol. 20, Ho. 4, 1992, 53-64. [2] E. Atanassov, Universal generalized nets. II,

AHSE

Review, Vol.

21 (1992), Ho. 1, 59-64. [3] E. Atanassov, L. Atanassova, E. Dimitrov, ski, H. Harinov and S. Petkov,

G. Gargov, I. Kazalar-

Generalized nets and expert

sys­

tems. Hethods of Operations Research, Vol. 59. , Proc. of the th Symposium on Operations Research of the

Gesellscbaft

thematiK, Okonomie und Operations Research, 1987, Passau,

12-

fur HaFrank­

furt a.H: Athenaeum, 1969, 301-310. [4] E. Atanassov, L. Atanassova, E. Dimitrov,

G. Gargov, I. Kazalar-

ski, H. Harinov and S. Petkov,

Generalized nets and expert

tems.

"Hetworks Information

II,

Proc of Int. Conf.

sys­

Processing

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

29

Systems", Sofia, May, 1988, Vol. 2, 54-67. [5] K. Atanassov, L. Atanassova, E. Dimitrov,

G. Gargov, I. Kazalar-

sKi, M. Marinov, S. Petkov and M, Stefanova-Pavlova, nets and expert Vol. 63.

systems. III.

Proc. of the 14-th

Generalized

Methods of

Operations

Research,

Symposium on

Operations

Research.

Ulm, Sept. 1989, 417-423. [6] K. Atanassov, L. Atanassova, E. Dimitrov, ski, M. Marinov and S. Petkov, tems. IV.

Proc. of the

G. Gargov, I. Kazalar-

Generalized nets and expert

XIX Spring

Conf. of

sys­

the Union of Bulg.

Math. , Sunny Beach, April 1990, 155-161. [7] L. Hadjyisky

and

K. Atanassov,

A generalized net, representing

the elements of one neuron network set, AMSE Review

Vol. 14, No.

4, 1990, 55-59. [8] L. Hadjyisky L. and K. Atanassov K. Theorem for representation of the neuronal networks by generalized nets, AMSE Review

Vol. 12,

No. 3, 1990, 47-54. [9] Handbook of operations research foundations and fundamentals Moder and

S. Elmaghraby,

(J.

Eds. ), Van Nostrand Reinhold Co. , New

York, 1978 (Vol. 1 and 2). [10] R. Ackoff,

The art of problem solving,

A Wiley-Interscience Pu­

blication, New York, 1978. [11] J. Kleijnen, Statistical techniques in simulation, M

Dekker INC,

New York, 1974 (Volss. 1 and 2). [12] P. Fishburn, Utility theory for decision making, J. Wiley & Sons, INC, New York, 1970. [13] H

Wagner, Principles of operations research, Prectice-Hall, Inc.

1969. [14] J. Kanter,

Management-oriented

management

information systems,

Prentice-Hall Inc. , Englewood Clitts, New Jersay, 1977. [15] R. Ackoff

and M

Sasieni,

Fundamentals of operations research,

John Wiley and Sons, Inc. , New York, 1968.

30

CHAFFER 1

[16] Atanassov K. , Open problems in the theory of generalized nets, Preprint IM-MKAIS-1-91, Sofia, 1991. [17] Atanassov E., Universal generalized nets, AHSE Review, Vol. 7 (1988), Ho. 4, 29-35.

HEM RESULTS IH THE THEORY OF GENERALIZED HETS

31

§ 1.2: ON THE REPRESENTATION OF GENERALIZED NET TRANSITIONS E7 A FIXED SET OF TRANSITIONS Favlin Gyurov Let

and Krassimir Atanassov

a fixed GN and a set S with elements - fixed graphical

structures of transitions be given. We snail describe an algorithm for a representation

of the transitions of the given GN E by the elements

of the set S. Thus we shall solve the problem 12 of App. 1: "To construct

an algorithm which for every set

of transition

graphical structures (obviously they must satisfy some conditions) and for every given GN constructs a GN for which (a) all its transitions have forms which are elements of the given set; (b) both nets have and

as a result

equal tokens with equal initial characteristics, of their functioning,

the equal tokens

receive

equal final characteristics. " The mentioned conditions for the form of

transition can be as

follows:

1 |LJ I i IS I + 1 Z€S Z (i. e. it is necessary that at least one transition of S exists which

contains at least two input places) and 2 |L"| i |S| + 1 Z€S Z (i. e. it is necessary -that at least one transition of contains at least two output places),

S

exists which

where IXI is the cardinality of

the set X. Let the above the set S = 1Z , Z 1 2

two conditions be

satisfied for the elements of

Z ] and let the GN s

o « E = <, , , <X, i, b>> A L 1 2 K K be given, where the elements of the set A of the GN's transitions have the forms

32

CHAPTER 1

2 = , 1 2 (see App. 1) and l e t

r = [L J , L", ||r

II]

( s e e App. 2 ) .

V, »

From

the Theorem for

the completeness of

(see § 5 of App. 1), it follows that every GH can

the GH

transitions

be represented by a

union, a composition and an iteration of its transitions, and therefo­ re it is sufficient to show a method for representation of an arbitra­ ry GH's transition and this can also be transformed in all other tran­ sitions. Hie new GH will have transitions with

S-fonns.

It is conveni­

ent that this GH be a GH with a global memory (see § 3 in App. 1). Let this new component contain

as parameters the lists of current capaci­

ties of the transition's arcs and of the numbers in the output places: for a transition

2

of the free location

these lists are:

H Z

= [L', Z

L", llm II], where m are the current capacities of the transition's Z i,j i, j arcs between places 1 and 1 ; and (<1, c(l)-c(l, TIME)>/l € L"], where i J Z c is a function giving the number of tokens in place

1 at the current

time-moment TIME. The GH transitions can be ordered according to

their prioriti­

es. Let a transition Z of GH E be given. Our algorithm is as follows: By

"//...//" we shall denote some

comments. A01. Order the input and output places of Z by their priorities, where the order in the first case is descending and in the second

case

in ascending series. A02. Check the inequality |pr Z I i ipr 2| 1 i 1

(1)

for every i (1 i i i sj, where pr X is the j-th projection of the j

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

33

n-dimensional set X (1 i J i n) and S - [Z , Z 1 If the set of the S-transitlons

Z1 2

s

which satisfy (1) is empty, 90

to A07. A03. Check the inequality ipr S 1 z ipr Z| ipr2S 1 1 z ipr2Z| 2 i 2

(2) (2)

for every i (l <, i i s). If the set of the S-transitions

which satisfy (2) is empty, go

to A06. K

A04. Determine this s-transition Z , which satisfies (1) and (2), for which K It

It K

ipr |pr Z 1 + Ipr Z I 1 2 is minimum

* A05. Mark the first Ipr Zl 1

input places of Z

with the corresponding

place's identifiers of the input places of Z and the first Ipr Zl 2 it

output places of Z with the corresponding place's identifiers of it

the output places of Z; (different) *

*

If

t

It

t

z 2

Z

z

Z = t , 1 1

z

=t , 2

- L") and define it

Z

Z Z z

identifiers - these places are

- L') V (L"

(L'

mark the other places of

Z

Z

with other

elements of the

set

34

CHAPTER i

r r

= rr K

+ r, r, Z Z

z z

where r = [L' « Z

- L', L" - L" , llfalsell] Z * Z Z

is a

p r e d i c a t i v e index

matrix, M M **

Z z

; : M M + M, M, Z Z

where M = [L'

z

matrix, D D

»» z

- L\ K

L"

Z

z

- L" , II 0 II] «

i s a natural number index

Z

=T T || D ,, V V ll l l ll//HH L L' ' - L')>>. L')>>. Z «« Z z z

z

z

z

GO t o A20. //Obviously !D

iff !D , Z

it

z because the tokens do not enter in the places of the set L' »

- h'. Z

z The method

of checking

the functioning of

a transition

structed in such a way is similar to the one shown below

and that

con­ is

why it will be emitted here. // R

A06. Mark the first Ipr Z| 1

input places of Z

with the corresponding

place's identifiers of the input places of Z, and the other input K

places of

Z

with other (different)

are elements of the set (L' «

- L'), Z

identifiers - these places

Go to A10.

Z A07. Check the inequality (2) and if it is not valid, go to

Aoa, else

« mark the first Ipr Zi output places of Z 1

with the corresponding

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

35

place's identifiers of the output places of Z, and the other outK

put places of Z

with other (different)

ces are elements of the set (L"

* z

identifiers - these pla­

- L"). Go to A08.

z

A08. Let Z ; Z. Let i ~- 1. 0 A09. Determine this transition of places whose number

is

S with minimal number of input pla-

greater than

|Z

1-1

or,

if such a

i-1 transition does not exist, the S-transltion with the maximm num­ ber of input places

(let it be

Z'). Construct a transition Z i i from a transition Z by removing its final Ipr Z'I - 1 from the i-1 1i number of places. Mark the first input place of Z' by y' (an 1 identifier

which is not found among the other GN's

i

identifiers)

and the other input places of Z' with the identifiers of the cori responding input places of Z The final output place of the i-1 transition Z' is marKed by y' (it is the first input place of i i-1 the transition Z' , if it exists). i-1

MarK all other output places

of Z' with other identifiers which are not found among the other i GN's identifiers: x' , x' i, 1 i, 2

x'

. Define i, IL' 1-1 Z' i

c(x' > - O, o, i, J j w <x' ) = 0 (for 1 i j i |L' I -1), L i, j Z' i c(y' ) = c(y' > = oo, > i-1 i

36

CHAPTER 1

n (y') = max |n (1) / (1 € pr 2') 8 (1 * y')]. L i L 2 i i If the -transition

Z

has no input places

(i. e. if the second

i case mentioned above is valid), is valid for Z,

then finish the procedure, and if (2)

determine the priorities

and capacities of the other

input places of Z'-transitions, in relation to the corresponding prio­ rities

and capacities of the places of

Z and

go to A12,

else go to

A10; else i:= i * 1 and execute A09 again. A10. Let Z = Z. Let i r 1. 0 All. Determine this transition of

S

which has the minimum

output places whose number is greater than |Z

number of

1-1

or, if such

S-transition with

the ""'nim

i-1 a transition does not exist,

the

number of output places (let it be Z"). Construct a transition Z i i from a transition Z by removing its final Ipr Z" I - 1 for the i-1 1i number of places. Hark the first output place of Z" by y" (an i identifier, which is not found among the other GH's

i identifiers)

and the other output places of Z" - with the identifiers of the i corresponding output places of Z . The final input place of the i-1 transition Z", is marked by y" (the first output place of the i i-1 transition Z" , if it exists). Hark all other input places of Z" i-1 i with other identifiers

which are not found among the

identifiers: x" , x" ,..., x" . Define i, 1 1,2 i, IL" |-1 Z" i c(x"

) = 0,

other GHJ s

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

37

f (x" ) |L" I I --1), IT T (x" ) = 0 (for ( f o r 1 i<. j j i i |L" 1), L Z' ii,J ,J Z' i i c(y" ) ) ^ r c(y") c(y") = to, c(y" to, i-1 i-l 1 ir (y") ir (y") = max [u [u (1) (1) / / (1 (I € pr pr Z') Z') & & (1 (1 * y')j. y')i. L ii L 22 ii ii If the transition

Z

has no output places

(i. e. if the second

i case

mentioned above is valid),

then finish the procedure, determine

the priorities and capacities of the other output places of Z"-transi­ tions,

in relation to

the corresponding priorities and capacities of

the places of Z and go to A12; else i:= i + 1 and execute All again. A12. Identify places y' and y" by the operator O (see § 5 in App. 1). 0 0 //See structures

Fig.

1. 6.

Nore that

by these

12 steps

the graphical

of the transitions which represent the given transition

Z

are determined. Below we shall determine the other components of the above con­ structed transitions. // A13. Let the number of the transitions of the first (left) group be n 1 and that of the second (right) group be n . Determine a new tlme2 step t' for which o t t' =

. n

+n 1

+1 2

//More generally, n 1

= max pr Y, Yes l

2 2

= max pr Y, Y€S 2 Y€S 2

n

and these values can be determined before the application gorithm. // A14. Determine the following time-components

of this al­

38

CHAPTER 1

Fig.

1. 6

NEW RESULTS IN THE THEORY OF GENERALIZES) NETS

39

Z' Z' i i Z t t :-. tt *+ ( (n n --i i). | . Vt' I(lt ii i is <. n n|,), 1 11 1 11 11 Z" i i ZZ t (n -- 11 + ii). (1 ii ii i.i nn ). t = tt + (n ) . tt' ' (1 ). I l l 2 I l l A15. Determine all elements of the M-components of

the transitions as

"eo". A16. Determine the transition types

of all transitions as

"v"

(i.e.

they will be activated when at least one token enters some of its places). A17. Define the characteristic function

$

in the different places as

follows $

gives as the current characteristic of the token the list of

y i the output Z-places in which the token mist enter, $ , $ , $ are not defined, x' x" y" i,j i, J i $ 1" i

coincides with § which is defined for the given GN. 1" i

Ala. Determine the transition condition predicates sition from the left group

of the

i-th tran­

4©

CHAPTER 1

x'

false

. . .

1' i, 1

I false iI I I false I

false

...

false

W

1' 1,2

I false | .

false .

... .

false

W

I false I| I I

false

...

false

W

= i

y

x'

1,2

y' iI r

...

x'

1,11 1,

1' 1, IL' 1-1 i, i-l Z' 1

i, IL" 1-1 Z' 1 false

i-l i-1

true

where UV [1' IV , 1 '' i 1, 1 1, 2

1' ) -.-- L' , 1, IL' i-l i, 1-1 Z Z' 1

and W

-. " ( 3 T : 1 £ T i

|L' l)((f(r ) = 1) & (m > 0) & Z' p, T p, T 1

0 ) ) " , T T where 1" is the index of the p-th row of the index matrix r . P 1 //The values of ID and of c(l", TIME) are located in the gloP, T T bal GN-memory. // Determine

the transition

sition from the right group:

condition predicates of the j-th tran­

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

y' y i i

x' i, i, 1 1

r = ri = i

x' x' i,2 i,2

I false | I I I I| false | false . | .

x' x'

1" i" J, 1 1 J,

1" i" ... j, 2 2 j,

ii-i -l

1" i" i, |L" 1-1 i, |L" i-l Z" Z" i i

false

false

. . .

false

false false . .

false false . .

. . . .. . . .

false false

|| false false false false false false

... ...

false false

ii

... ...

w

ii, , IL' IL' | -1-1 1 II Z" II ii II

y" y"

41

w

w

w

iI

where U (1" (1" , , 11 "" J J jj, , 11 jj, , 22

1" 1" ) ) = L" |L" I -1-1 j j, , |L" l ZZ Z" Z" J J

and

W = "There exists at least one identifier of the place of a tran­ sition with a bigger number,

which is an element of the set

of the last toKen characteristic", W" - "The identifier

of the output

place is

an element

of the

set - the last token characteristic", A19. Define: n {Z') = if (Z") = n (Z) for every Z' and Z" - transitions A i A J A i J from the left and from the right groups, respectively. A20. Check the list of the transitions of GN E. If all transitions are processed - end of the procedure; else - go to A01. it

Let the newly constructed GN

be marked by

E . From the above K

construction, it follows that all transitions of the set S.

Moreover, the last characteristics

E

are elements

of

which the tokens will

receive in Z"-transltion places will coincide with the characteristics J which the tokens will receive in the corresponding Z-places.

42

CHAPTER 1

Therefore the following theorem is valid. THBQRHJ: For every set of transition tisfy the inequalities

graphical structures,

(1) and (2)

and for every

which sa­ given GH,

there exists a GN for vtfiich - all its

transitions

have forms nfliich are elements

of the

given set, - both nets have equal tokens with equal initial characteris­ tics and,

as a result of their functioning,

the equal to­

kens receive equal final characteristics. This paper is based on [1].

HEFEREHCE: [1] F. Gyurov and X. Atanassov,

On the representation

net transitions by fixed set of transitions, 91, Sofia, 1991.

of generalized

Preprint IH-HFAIS-2-

HEM RESULTS IN THE THEORY OF GENERALIZED NETS

43

§ 1. 3: TWO HEW CONSERVATIVE EXTENSIONS OF GENERALIZED NETS Rumen S. Christov

Some problems

and

Krassimir T. Atanassov

related to the extensions of the concept

"gene­

ralized net" (GH) were generated during the development of the program realization of GNs. Below we discuss two new types

of nets

and prove

that they are conservative extensions of the ordinary GHs. Let the object o « <, , <X, i, A L 1 a E K be given

and

let it have the following property:

place of the set

b>; L'>

when in

a certain

L' c L there exist at least two of its tokens, which

are generated by one

token "predecessor"

independently from the fact

that (a) there can exist other "kindred" tokens somewhere (i.e. tokens with the same "predecessor"), (b) the tokens can be generated

in different moments and

from diffe­

rent, but kindred, "predecessors", then these tokens are united in one token by analogy with the procedu­ re from

§ 8 in App. 1. The new token

will also be a kindred token of

the tokens which are kindred tokens of its generated ones. Let us call this new net

"a GH allowing

a particular

token's

uniting" (GH-APTU). THEOREM 1.3. 1: I

c I. GH-APTU

Proof: Obviously, every GH-APTU for which

L' = $,

is not defined
i. e.

for which

X

i- £.

LJ Let

GH-APTU a GH-APTU E be given. Initially, we construct an ordinary

GH

E

with

the same graphical structure. After this, for each output place 1 € L' of transition Z of GH-APTU E (let 1 be a fixed place), we construct a new transition Z which corresponds to 1 (this transition contains 1

44

CHAPTER 1

place

1 as an input place).

This new transition

(see Fig, 1.7) has

a transition condition with the form 1 rr

= 11 1

II 11

Z II 1 1 II 2 1 2 1

Z 11 1

W W W W 11 22 W W

ffalse alse

44

ZZ 1 2 W W 3 W W 5

where Z = "the token has no kindred tokens in places 1 and 1 "; 2

W 1 W

Z = "in place 1 there is already a kindred token of the present 2 2

to-

ken"; = i W

W 3

1

8 1W, 2

W = "the token has no kindred tokens in place 1 and it has received 4 as a current characteristic the characteristics of the last toZ ken in the place 1 " (this is described below in relation to the 1 characteristic function), W

= nW . 5 4 The following inequalities for the new places are valid -Z ZZ n (1) (1) > n (1 (1 ) ) > TII (1 (1 ) ) L L II L 2

and Z n (1) T (1) > T *I (1 (1 )• ). L L 2 Let the GH-characteristic function that in place

Z 1 2

the token receives

as

$ be defined in such new characteristic all

a way the

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

46

Fig. 1. 7 Z characteristics, which the token, last entered place 1 , has received 1 Z after the time-moment when it has split with the token from place 1 , 2 Z or with its corresponding "predecessor". In place 1 the tokens do not 1 receive new characteristics.

In place

1 the tokens receive the same

characteristics as the tokens of GN-APTU E in place 1. For transition Z from E (we denote it by Z ), let E L Z

= (1/1 € L' D pr Z ). 2 E

Following the above algorithm we can construct the set

of transitions

of E (Z /l € L ), and to transition Z of E we Juxtapose the transition 1 Z Z : Z V U Z. 1€L 1 Z From this construction it is obvious that the transition Z from E will function in the same way as the transition

Z,

By analogy with

the proof of Theorem 5. 3. 1 [*] (see | 5 in App. l) we can prove

that

46

CHAPTER 5

the GHs

E

and E

are equivalent in the results of their functioning.

Therefore E represents GH-APTU E and hence J c J

.0

GH-APTU Hie other extension of the concept GH is related to the charac­ teristic

functions of the nets.

How they are

related to the tokens.

Tokens receive new values from the characteristic functions associated with the places which the tokens enter.

These functions may be depen­

dent on the entire information in the net. On the other band, the pla­ ces are also influenced by

the processes in the nets, but they do not

have their characteristics. It can be obtained by defining the charac­ teristic functions which will give characteristics to the places. Let the net E E

= =

<
i,

n , n , c, f, e , e >, nA, nL, c, f, e1, e2>, A b»

L

be given, for which

1

$

o « <E, n , e >, , K E <E, n , e >, ,

2

K

<x, x, $, <x, x, $,

E

is a function determining

the places' current

characteristics. For this net the places have initial

characteristics

which are elements of the set X. we shall call this net a GH with pla­ ces' characteristic functions (GH-PCF), The proof of the following assertion is easy. THEOREM 1. 3.5: £ : J, GH-PCF Every GH with a global memory presented

by an ordinary

GH.

characteristic functions (related at

different moments

(see § 3 in App. 1)

From the fact

that all

can be re­

values of all

to tokens or places) are calculated

of the GN's functioning,

it follows that these

results can be remembered in the GH "memory" and therefore the GH "me­ mory" can represent them Therefore al 1 GH-PCF can be represented by a GH with a global memory. On the other band, this type of extensions of

NEW RESULTS IN THE THEORY OF GENERALIZED NETS

47

a GN can be represented by an ordinary GN, from which, the validity of the Theorem follows,

because the opposite is obvious:

every ordinary

GN is a GN-PCF without the specific components. 0 From the construction

of the above proof

follows the validity

of THEOREM 1. 3. 3: Every ordinary GN (or a GN-PCF) can be represented by a reduced GN with

a global memory but without the set of

initial characteristics and ons.

the characteristic functi­

Chapter

2:

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

In this chapter different results are included ted to

the application of

the GHs in some areas

applied mathematics, artificial intelligence and illustrations

ntfiicb are rela­

of science

scientometrix.

of the possible applications of GHs in science

ven. Four of the papers

<§ 2. 1, §2.2, § 2.4

and § 2.6)

Thus

are gi­

are revised

versions of old papers, and the other four contain new results.

46

such as

SCHE APPLICATIONS OP GENERALIZED HEXS IB SCIENCE

§ 2. 1:

49

GENERALIZED NETS REPRESENTING THE ELEMENTS OF NEURON NETWORKS Ljubomir HadjyisKy

and Krassimir Atanassov

The class of Neuron BetworKs

(NNs)

represented with GNs is a variety of the

(see [1-3])

which will be

"additive model" [4).

This

class of HNs is described by the equation Y(t 4 at) = g(I
T B y (t)> € R B

(R is the set of all real numbers,

T <. . . >

is the transposed vector

of the indicated one. ] is the state of the BB at the time-moment t; (e) y (t) is the state of the neuron "i" at the time-moment t; i B x H (f) T € R

i s a matrix whose elements

T (1 i i, i, j

synaptic strength of the connection between the neuron

j i B) are t h e " j " and the

neuron "i"; I(t) = =
(g)

T B I (t)> (t)> € € R R I B B

is the potential of the somna obtained by integration of the separa­ te input impacts; (i) (i i

B I (t) = B2 T .y (t) Ii |t| •■ j=l I I ii J .JJ (t) i j=l ii J J

i <■ H ) ;

T B (j) gd(t)) = € R 1 2 B is an (B x 1)-vector function determining the state of

the neurons.

50

CHAPTER 2

N H X H (R is a linear N-dimensional vector space, R is a linear (NxH) -dimensional vector space.) y (t) describes the state of the neuron "i". Ibis state can i exited or not and, according to this

y (t), accepts i

be

corresponding 1y

the values 1 or 0. The determination of g (I (t)) is realised by the following rei i 1ations:

4

(\

, if I (t) > e

i

7 (t) = g (I (t)) = 1 0 , if y i (t) = g i (ii (t)) = { o , if i i i y (t), if y (t), if where 6

I (t) ii (t) Ii (t) I (t)

'

< G , < ei , = 6i = 6

is the threshold value for the neuron "i". i

This class of HHs has the possibility of self-training, sed by changes of the synaptic streingths T

reali­

or the threshold values i, j

6 . In case some neurons are dead,

the network can reorganize

itself

i and the other neurons will accept the functions of the dead ones.

The

representation with GHs includes this property as well. Let the GH and

E

E

be composed of two

graphical incoherent GHs, E

(see Fig. 2.1). GB E is a modelled 2 1

1 I(t)

somatical potential

and a determined state of the neurons Tf(t + at) by g(I(t)). GH E

per2

forms self-training and reorganization When a neuron is dead. E

The

GH

is composed of nine places, three transitions and H tokens, where H 1

is

the number of neurons in the network.

1 is c|l ] -- I i i

for

i i i i

9,

The capacity of every place

i. e. at some time-moment all tokens

9C*E AFFLICATICHS OF GEHERALIZED HETS IH SCIBHCE

can fall into one of the places.

It is convenient

51

to use an extended

GH with a global nEmory (see § 3 in App. 1). In this case, in the com-

Fifi. 2.1

5a

CHAPTER 2

ponent B different data sets about the net-topology IT J, i, j

the SOEOB-

tical potential of neurons (I (t)l, their threshold values 16 }, their i i state ly (t)] can be disposed of as follows ii JJ (1) (1) JJ (2) <2) .. .. .. N i i i i (a) X+ x+ i=l T T i=l ii II T T .. .. .. I I JJ ((l),i l),i JJ ((2), 2 ) , ii II ii ii where

H £+ marks the sum of i= l

number of the neuron

"i",

N r

JJ ((r r ) ) i i ii

, T T

JJ (|r],i r ],i ii ii

index matrices (see App. 2), i

is the

is the number of the neurons which are i

connected with neuron "i", JJ (k) / 1 i k i r 1 is the set of the numi i bers of the neurons which are connected with the neuron "i", for 1 4 JJ <, r , T is the synaptic strength of p-th connections, i J (JJ), i i ii JJ B
(d) (d)

H JJt t ii=l =l

ii

ii i i e ,, II

ii ii

i i l |l j |t] (t) . || ii

The relation among

places

is given

by the conditions of

index matrices for the corresponding transitions: 1

r

=

1

1

22

33

1

I W 1 1 1

W 2

V

1

I W 5 | 1

W 2

W 3

I W 1 1

W 2

W

1 1

7

44 3

3

,

the

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

where 1

- "y (t) = 1" & nW , i 3

2

- "y (t) = 0" & 1W , 1 3

W W

- "T

W 3

= 0"; a 1 5 r

= 2

1 6

1 l W 2 1 4

W , 5

1 I W 6 | 4

W 5

where W =W v W , 4 2 3 W 5

=W ; 1 1 7 7 r

= 3

1 8 8

1

1 I W 3 | 6

W

W

1 I W 9 1 1 6 9 6

W

9 9 ,

7

8

7 7

W 9 9

where W

=W 6

1

7

zW, 2

W W

V W , 3

= "no connection exists", 8

w

= "C - O", 9

where (i, if a neuron a is a live, = V (t) = \ a. a I O, if a neuron a is a dead,

T

V (t) is the function determining the physical existence of the neua ron a.

53

54

CHAPTER 2 $

The characteristic functions

(1 i i i 9)

are not defined.

i Hote, that if we use an ordinary GH, B,

i.e. a GN

without global memory

then a part of the characteristic functions will exist.

Hence the

function i is not defined. The time-moments

for firing

and for activity duration

of the

transitions are t

1 1

= t , o

2 t = - tJ , 1 t

3 1

= t , o

1 t

2

- 00

,

2 t = t" , 2 3 t

2 2

= 0D 00 ,

where where t (= T) is the initial time-moment of the HH and hence of to (= T) is the initial time-moment of the HH and hence of o

(

t

+ t

i if t = t

and there are no tokens in the

place 1 , 6

tJ =

+ t

E 2

+ t", otherwise,

o n o place 1 , ] and is the time at which |I (t) 6 i t + t + t", otherwise, ted, calc E t is the time at2which the unique token tE is the time at which |I (t) ] and calc i 2 ted, t is the time at which the unique token E 2 t calc

the GH E, the GH E,

|y (t + at)] will be calculai

of the GH E does one cir|y (t + at)] 2will be calculai of the GH

E 2

does one cir-

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

55

cle in E , 2 t" is the time at which N tokens transfer from input places

t n

to output

places of the transition Z , 2 is the time at which a token can be transferred from the transition Z 1

to the transition Z . 2 The calculation of (I (t)) and (y (t + 3t)I begins at the timei i

moment t'. When t = t , the values o

(y (t)J i

of some neurons which are

inputs in the NN can be changed and this will be an external impact to the NN. The GN E

is composed

of five places, two transitions

and one

2 token. The capacity of every place is 1. In this GN, self-training and reorganization

are accomplished.

Bearing in mind that the previous s states of the NN are remembered as a matrix (y ) (1 i s i n), where n i

is the number of this state.

They are selected by determined criteria

in the component B of the GN E. The relations among the places are given by the conditions: 1 11 11

r = 4

where - "F - 1",

W 10 w 11

= nW , 10

1 12 12

l 1 i w 10 I 10 lO

W w 11

1 I W 13 I 10

W 11

l I w 14 I lO

w 11

56

CHAPTER 2

1 13 13 r

= 5 5

1 14 14

1 | W 13 I 12 13 I 12

W 13 13

1 I W 14 12 14 I I 12

W 13 13

where W

= "F = 1", 12 - IV

W

13

,

12

where the parameters R and F have the following values: 1, if C d, old

= C, d

d, old


E= 0, if C where C

is the number of dead neurons before the last cycle, d, old

C d

is the current number of dead neurons; f1, if non-self-training, F ---- \ i F \ 0, if self-training. The values C

, C and F d, old d

component B of the GH E.

are defined as

parameters for the

The value of F is defined at the initial ti­

me-moment. The characteristic functions $

(10 & i i 14) are not defii

ned. The time-moments for firing and activity duration of the transi­ tions Z and Z are 4 5 4 t -- t 1 calc 4 t =t 2 tr 5 t = t (= T) , 1 o

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

57

5 tt

= = tt

2

tr tr

where t

tr

is the time when one token transfers from an input

place

to an

output place of one transition, tt = tt + tt ,, E E rr r 2 44 5 ft r 4

II reor // tt ,, iif f R = 0,0,

[

ft /t tt

r

,, iif f R R -: 1,1,

-- . 5

t r 4

\ tr tr

,, iif f P -. 1,I,

self self tt ,, iif f P = 0,0, E E * * 2 2 is the time when a token is transferred throughout the transition

Z, 4 t r

is the time when a token is transferred throughout the transition 5

Z, 5 reor t is the time when the MS reorganizes when a neuron is dead, E 2 self t is the time when the NN is self-trained. E 2 The calculation of new characteristics of the NN when the net­ work has been reorganized begins at the time-moment t

and when i t calc

58

CHAPTER 2

has been self-trained, at the time-moment t + t . calc r 4 Let T be the starting moment at which GH t

be toe running time

E begins functioning,

of the model, given by a time-variable, diffe­

rent from the time-variables of the GH

E, defined for the component B

of the GH. At the moment T, all tokens are in the place 1 . The transition 1 1 1 Z is fired (t = T, t =
the?

Heurons of

1 are transferred in places 1 and 1 (transition Z is fired). 3 7 8 3

Heu-

rons which change their state from non-excited to excited or dead fall in place 1 ; in place 1 , neurons which keep their 7 8 ve. The conditions for tokens transfer from in in 1 , 8 neurons

state and are ali-

1 in 1 , or from 1 aga8 7 8

are the same as the ones from 1 to 1 or 1 . Only excited 3 8 7

can change

the state

this case tokens (neurons)

of other neurons from the network.

transfer through

the transition

In

Z . The 2

change in the neuron state will be described in detail. At the starting-moment, place 1 ; 6

the transition Z

when t = T = t , o

there

are tokens in

is fired at the time-moment t 2

+ t , i. e. o n

when a token appears in place 1 and the duration of the fired transi2

SC*E APPLICATIONS OF GENERALIZED NETS IN SCIENCE

tion is t".

At the

time-moment

59

t + t" the calculation of o

(I (t)), i

(y (t)) and E begins as follows: let a token with a number a be trani 2 sferred through Z . The somatica1 potentials 2

I of J
the neurons

connected with neuron a will be I (t) -. T .y (t), J (k) J (k),a a a a with K = 1, 2

r. a

I (t) is the change of the potential of neuron J (k) and it J (k) a a is added to

the old value of this potential defined as

an element of

the component B. In the same way, for all other tokens passing through Z , the potentials of the neurons are determined in the NN. 2

When this

calculation is finished for all tokens, potentials I (l < a i N) a

are

determined completely. Then the calculation continues with the functi­ on g (I (t)) for 1 £ a £ N and the new states ly (t + at)! are detera a i mined. The running time t

is equal to t .At this moment the trancalc

sition Z of E is fired and 4 2

the token moves to place 1 or 11

1 . If 12

the token falls in place 1 , a reorganization is not necessary, 11 the number 1 , 12

of neurons does not decrease.

a reorganization is necessary in

If a token

falls

i. e.

in place

the data of the neurons in the

NN and the connections among them At the time t = t + t + t", calc r 4 the reorganization is completed and the transition

Z

is fired. 5

The

60

CHAPTER 2

token falls in place 1 or 1 , where the corresponding self-training \1 13 is

either performed

transitions Z

or not at the time t . When GH r 5

and Z 1

E

is working, 2

are fired and the excited tokens (neurons)

are

3

moved correspondingly from place 1 to place 1 , and 1 . The dead neufl 7 2 rons are moved correspondingly from place 1 to place 8

1 and 1 . When 7 4

the running time t is equal to the following: t = t + t + t", calc E 2 the transition Z is fired and the running time 2 t . So excited tokens are removed from places 0

t

receives the value

1 and 2

1 to place 1 ; 6 6

the non excited and dead ones are entered in to place 1 . 5 the process continues. If all tokens fall in place 1 8 the not excited

In this way

this means that

neurons in the network are absent and al 1

tokens are

entered into place 1 and the GH E stops functioning. 9 By the representation of HNs with GUs, the distribution of neu­ rons by their functional state is obtained. This allows a better visu­ alization during the UN

functioning, because only excited neurons ta­

ke part in the corresponding operations. On the other hand, it is pos­ sible

that some of

the operations for the neuron's

data change

are

performed simultaneously. These properties obtained by

the representation with

GH

as a

possibility for an external impact of neuron characteristics made pos­ sible

the effective modelling

with HHs

with processing of information.

»

of real processes

connected

S O E APPLICATIONS OF GENERALIZED NETS IN SCIENCE

61

Below we shall define a new type of NNs and construct

a GN

E

which will represent their functioning and the results of their worK. Each

new NN has the same elements (neurons) as the NNs from

above. The functions, which determine the functioning,

the state

and

the potential (Y, I, g, etc. ) of the new NNs are the same as the first NNs, described by the GN above. The basic difference between the two types of networks

is the

possibility of new neurons being born in the second type of NNs. These neurons possess the same functional possibilities as the others in the network. In the second type of NNs new functions exist

(which are de­

ft fined in the global memory

B

of the GN E ):

(a) V
a ) - the set of born neurons; V (t) 1 (c) V (N, t) = N + V (t) - the new neuron number for NN (N is the num3

1

ber of neurons in the NN before time-moment t; (d) V (x, t) - T U T , where 1 i i, j, x S V (N, t), x € V (t) 4 x, J i, x 3 2 and T , T are as above; x, j i, x (e) V (x, t) = 6 for x 6 V for x e V (t) (for g and I see above). 6 x x 2 x x The functions V

and V 1

to the network

are stochastic or they can

be

related

4

parameters.

strongly connected with V

The functions

V , 2

V , 3

V

and 5

V

are 6

and its type. 1

At time-moment t of being borning,

every new neuron

x 6 V (t) 2

62

CHAPTER 2

has a zero potential

(I (t) = 0) and a zero state (Y (t) = 0), where z x

Y is defined by g X

and I as from above. X X Q How the function g(I(t)} e E , where Q is the number of the na­ tural numbers.

* The GH E has the form of Fig. 2.8 ons which are identical in both nets E

(cf. Fig. 2.1). All notati-

and E

are marked identically.

a The new places in E

are m , m , m and m ; the new transitions are Z 1 2 3 4 6

and Z (a t r a n s i t i o n Z of GH E does not e x i s t h e r e ) . 7 5

«

Z 7

r e a l i z e s the c o n n e c t i o n between t h e subnets E 1

analogous t o E

and E 1

which

are

>

The t r a n s i t i o n c o n d i t i o n s i n t h e GN E

= 1

and E , 2

i n the above GN E. 2

r

The t r a n s i t i o n

«

are a s f o l l o w s :

1 2

1 3

1

1

I W 1 1 1

W 2

W 3

1

I 5 1

W 1

W 2

W

1 I W 6 | 1

W 2

W 3

m I 4 1 4 1

W 2 2

W

W 1 1

4

3

3 3

where W = "y (TIME) : I" 8 lV : 1 i 3 W = "y (TIME) = 0" S lV ; 2 i 3 W = "6 = 0 " (for 6 s e e above). 3 a a The t r a n s i t i o n c o n d i t i o n s r , r and r are t h e same 2 3 4

a s above

SCME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

53

t

Fig. 2. 2 The transition Z

does not exist and hence the same is valid for r . 5

a

64

CHAPTER 2

rr 6 6

m ID 11

m m 2 2

]] || 11 || 11

V V

W W 14 14

15 15

1 II 12 I 12 I

W W 14 14

W W 15

Kfoere W 14

= "V (TIME) > 0"; 1

15

= "V (TIME) = 0". 1

W

x In place

m

the control token (of the second subnet of

GH

E

1 which corresponds to the GH E

from above),

receives a characteristic

2 "V (TIHE)", vfoere TIME i s the current time-moment of the GH-model. The 1 condition m 3

r 7

m

l 4

l 13

14

m I II

W 16

W 16

W 17

V

= m I 2 I

W 16

W 16

W 17

W

m I 3 1 3 1

W 16 16

W 16 16

W 17 17

W

18 18 18 18

where W = "x > 0"; 16 last W = "x = 0" 8 W ; 17 last 12 W - "x - 0 " ft W 18 last 13 (W and W are d e f i n e d above). 12 13 In p l a c e

m 3

the token r e c e i v e s a s c h a r a c t e r i s t i c

"x

- 1" last

and i n p l a c e m : "<x, V (x, TIHE), V (x, TIHE), V (x, TIHE), I (TIHE), 4 4 5 6 x

SCWE APPLICATIONS OF GENERALIZED NETS IN SCIENCE

65

Y (TIME)>", where x € V (IDE). x 2 All components In the GN global memory

are changed to new ones

for which N: = N + V (TIME) at the tune-moment TIME. 1 In this GN, the boundary of N is

2 and N

is a function of the

global model-time. Here the numbers of new connections between neurons are added to the elements r , if they are generated for the corresponi ding neurons. In this GN t E

= t + t + t , r r r 2 4 6 7

where t is defined above, r 4 / t

6

t

, if V (TIME) > 0

, otherwise (see above),

I tr tr where t W

is the time for the calculation of the function

W 1

and

1 t t r r

M M

7

= (TIME), t =V V (TIME), t 1 1

+ t , , + t r r 5

K

where t is the time for the calculation of the characteristic toKen in place m (for t see above). 4 r 5

of the

» The GN E

functions in a similar way as GN E. The tokens of the

NN enter E* in place 1 . These neurons which are generated at the time 1 of the functioning of E* GN E

appear in place

m . A control token 4

from the above transfers from transitions 2

of the

Z , Z and Z and is 4 6 7

"a mother" of the born neurons. In the other places

it functions only

66

CHAPTER 2

as a control token for the modelling process.

« The place capacities of GH E The other models

are infinite.

for functioning of HHs

can be represented by

similar GHs. The graphical structures of these GHs will be the same or simpler than the above ones. For example,

the GH which represents the

HH-back propagation method [5] will not have the transition

Z

from 3

the

first GH

functioning form

described above.

of HHs

On the other

band some ideas for the

(e.g. [6-9]) will force

some corrections in the

of the transition condition predicates

tions in this GH.

Finally,

and characteristic func­

some relations between the Petri nets and

some types of the HHs are discussed in [10-12]. This paper is based on [13, 14]. REFERENCES: [1] Jun-Vtei Wong,

Recognition of

general patterns using neural net­

works, Biol. Cybern. , Vol. 58 (1988), Ho. 6, 361-372. [2] H. Cottrell, Stability and atractivity in associative memory net­ works, Biol. Cybern., Vol. 58 (1988), Ho. 2, 129-139. [3] A. Fukushima,

A historical

neural network model for associative

memory. Biol. Cybern., Vol. 50, 1984, Ho. 2, 105-113. [4] A. Guez, V. Protopopsescu and J. Brahnen, On the stability stora­ ge capacity and design of continuous neural networks, Trans. SMC, Vol. 18 (1988), Ho. 1, 80-90. [5] B. Rumelhart and J. HcClellond,

Parallel distributed processing,

explorations in the microstructure of cognitron, HIT Press,

Cam­

bridge, Mass. , 1986. [6] J. Hopfield,

Heural networks phisical systems with emergent col­

lective computational abilities, 2554-2558.

Proc.

Hat.

Acad. Sci. , 1982,

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

[7] T. Kohonen,

Self-organization and associative memory,

67

Springer-

Verlag, Berlin, 1984. [8] L. Hadjyisky, Parallel processing

in frequency

dependent neural

network, Parallel and Distributed Processing, Elsevier Sci. Publ. , 1991, 59-74. [9] C. Lee, Intelligent

control based

theory. Proc, of Internat. Conf.

on fuzzy logic and neural net

on Fuzzy Logic

and Neural Net­

works, Japan, 1990, 759-764. [10] A. Negro, R. Tagliaferri and S. Tagliaferri, Some remarks on Petri nets and neuron networks. Proc. of the IASTE3) Int. Symp. App­ lied Informatics,

Grindelwald,

(M Hamza, Ed. ),

Anaheim,

Acta

Press, 1987, 28-29. [11] T. Kristian, J. Pavlasek and P. Sapaty,

Modeling of neuronal-ac-

tivity using modified Petri nets, Physiologia-Bohemoslovaca, Vol. 36, No. 6, 1987, 541-551. [12] 14 Zargham,

Parallel processing neural networks,

Proceedings of

the First Annual Meeting of the nternational Neural Network Soci­ ety, 1988, Boston; Neural Networks Vol. 1, No. 1, 1988, 558. [13] L. Hadjyisky and K. Atanassov,

Theorem for representation of the

neuronal networks by generalized nets. AMSE Review, Vol. 12, No. 3, 1990, 47-54. [14] L. Hadjyisky and K. Atanassov,

A generalized net,

representing

the elements of one neuron network set. AMSE Review Vol. 14, No, 4, 1990, 55-59.

68

CHAPTER 2

§ 2. 2: GENERALIZED HETS REPRESENTING THE TRAVELLING SALESMAN FROELEM Krassimir T. Atanassov

The Generalized Nets (GNs) have been used as a means for model­ ling real processes and theoretical problems.

For example the process

of solution of the transportation problem was described using a GH. Initially, ire shall construct a GH the process of

(see Fig. 2.3),

solution of a variant of

problem (see e.g. [1-7]).

After this,

describing

the Travelling Salesman (TS)

we shall define the TS problem

in a more extended form. The temporal components of this GN will be

o « T = 0, t = 0 , t =

1. The GN E consists of two subnets E 1 has only one transition Z

GH

"a formal description of the graph of connections

The predicate

one) is "true".

2

1

between points which must be visited by the TS exists in it. The transition Z is activated at every time-moment with 1 ration 1.

E

with one place 1 . One token a with an ini1

tial characteristic

and E . The second 2

of the transition condition

(there is

Toe token a receives, during its transfer from

a duonly 1 to 1

a 1 , the characteristics x , which we shall describe below. 1 i The token p enters GH E in place 1

with an

initial characte-

2 ristic "a name of vertex, where the TS has been initially". The transition condition

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

r 2 2

=

1 3 3

1 4 4

1

1 I 2 2 I I

W

W

W 3 3

1 I I 3 I 3 I

W

1 1

2 2 W

1 1

1 I W I 1 11 5 |

69

5 5

W 2 2

3 3

2

W 3

W

where W

= "From the vertex of the graph, described as

an initial characte-

1 ristic of a,

where

a

is the TS,

more than one path goes

out

which is not passed by the TS", W

= "From the vertex of the graph, described as an

initial characte-

2 ristic of a,

where

a

is the TS,

at least

one path goes out

which is not passed by the TS", W

= "From the vertex of the graph,

described as an initial characte-

3 ristic of or, where a is the TS,

no path goes out

which

is not

passed by the TS". If a token goes in the place 1

(i. e. the predicate W

5 lid), vertex,

then it receives as

the next characteristic:

where the TS is going",

is va-

3 "<"name of a new

"length of the path between the

old

and the present vertices", "cannon length of the TS transfer")-". "The length" can be changed with other coefficient. If the predicate W is also valid, each token from i splits into two tokens. The first enters the place 1

1 or 1 2 5

and the second,

the place 1 without a characteristic. 3 Thus all tokens which determine the different directions of the TS-transfer are collected in the place 1 . 5 When some of these tokens do not satisfy

the predicates W

and 1

70

CHAPTER 2

Fig. 2.3 W , the token enters the place 1 without a characteristic. 3 4 The condition 1 6 r 3

1 I W 4 | 4

1 7 W 5

where W = "There are no untravelled vertices in the graph described 4 initial characteristic of the token a " V

- "There are no untravelled vertices in the graph

as an

described as

an

5 initial characteristic of the token a,

but there is no

path to

it without repetition of the route". Each one of the tokens mon

length

of the path

which enters place 1 receives the coro6

as the current characteristic.

The

tokens,

SOME APPLICATIONS OF GENERALIZED HETS IN SCIENCE

nftuch enter place

1 leave tbe net with 7

71

a final characteristic

"the

token does not pass all graph vertices". the transition condition 1 8 8 r 4 4

1 | W 6 1 1 6 6 6 1 I a i 8 I

1 9 9 W 7 7

W 8

W 9

where W 6

6 a = (" £ c(l , TIHE) > 1" 8 " c(l , TIHE) = 0") v "x > x "; TIHE last i=3 i 8

W

6 a = " I c(l , TIHE) + c(l , TIHE) = 1" v "x > x "; 7 i=3 i 8 TIHE last

W

a = "x <, x "; 8 TIHE last

W 9

6 a = " I c(l , TIHE) = 0" v "X > X i=3 i TIHE last

where x

is the last characteristic of the corresponding token from last

a 1 and x is tbe current characteristic of tbe token a and 6 TIHE is the number of tokens in the place 1 at tbe time-moment ct racteristic x TIHE ristic of tbe token

c(l, t)

t. The ena­

is determined by the value of tbe current charactewhich is in

tbe place

1 . 8

In some time-moments

a x can be "0". TIHE Tbe classical TS problem

can be described by sush a GN.

problem can be generalized in some directions

This

(see e.g. [1-7]). Below

we shall introduce its new generalization and construct a GN represen­ ting one of them.

72

CHAPTER 2

Let a time-scale be fixed and let T

and T i

final time-moments of the process, which possible that T

be the initial and f

we shall model below.

It is

= co. f

Let

T = (T , T 1

T ] be a set of travelling salesmen (TSs; 2

which is different from TS

t as an abriviation of

"a travelling sales­

man") . Let C = |C , C C J be the set of al 1 products which the 1 2 c TSs delivered. Let V = IV , V V ] be the set of built-up areas 1 2 v (the graph-vertices] which can be visited by the TSs. Let us use everywhere the following indices: - i € 11, 2

t],

- j € II, 2

si,

- k 6 |1, 2

cl,

- TIHE € [T , T ]. i

f

Below we shall define some functions related to the elements of the above sets which will determine the choices of the TSs' routes: - »
C

1 being the set of the different i, c i

types of products, which the t-th TS delivered at the current moment TIHE, where t u »(T , TIHE) c C i=l i (We assume the possibility of absence of some products from the TSs' baggage at a given moment.); - Q(T , C , TIHE) is the quantity of the j-th i j i-th TS at the moment TIHE;

product

which has the

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

- P(T , C , TIME) i J

is the price of

73

the j-th product which the i-th TS

has at the moment TIME; - L(T , C , TIME) i J

is the

limit of fitness of the

j-th product which

the i-th TS has at the moment TIME; - M(V , TIME) = , where K K k, l k, 2 k, s k, J of the

different necessary

i 0 is the quantity

products in the built-up area V

at mo­ le

merit TIME; - D(V , V ) is the time for transfer from vertex K K' vertex V

V to its neighbour k

, the equality k' d(V , V ) = d(V , V ) k k' kJ k

is not necessary; - E(V , V , TIME) k k'

is the price of the transfer from vertex V to its k

neighbour vertex V

at the moment TIME. k'

- G(V , V , I D C ) € [0, i) k k' from a vertex V

determines the possibility

to its neighbour vertex k

V

of a

tranfer

at the moment

TIME.

k'

Thus, in the frames of a given time-interval,

the paths can be for­

bidden for transfers. - R(T , V , TIME) is the value of the gain of T i k i

in V . k

The value of

this function can be determined e. g. as follows: R(T , V , TIME) = I Q(T , C , TIME). P(T , C , TIME). i k C €*(T .TIME) i j i k J i sg(L(T , C , TIME)), i j when i

K k, j

E Q(T , C , TIME). C €V(T ,TIME) i j j i

74

CHAPTER 2

Analogously, the equality can be defined in the case when X < £ Q(T , C , TIME). k, j C €<MT , TIHE) i j j i - H(T , V , TIHE) is the time of staying of T in V after time-moment i k i k TIME. Based on these definitions, we shall formulate an the Travelling Salesman Problem (ETSP).

Extension of

We shall also formulate other

problems. The ETSP will be denoted as ETSP1. Trace out the optimal gain") route of the which

(according to a

criterion HG:

t-th TS among the independent TSs

must move in the built-up areas

V , V 1 2

V , v

"maximum

T , T 1 2

T t

carrying some

of the products C , C , . . . , C . 1 2 c THEOREM 1: There is a GH in the frames of which the ETSP1 is solved. Proof:

Let the GH from

Fig. 2.4 be constructed. It has the following

transition conditions: 1 3

r

= 1

1 I true 1 I I 1 | false 2 I 1

1

1 6

14

false

false

false

false

true

false

3

I 1

W 1

2

1 9

|I 1

W 1

2

1 I false 11 I

1 4

W

false

W 3

W

false

W 3

false

true

false

where W

= "there exists

at least one more path along which

the token

1 move"; W

= "there exists a path along which the token can be moved"; 2

can

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

Fig.

2.4

W = iW ; 3 2 5

r

= 1 I 2 4 1

W 4

where

a W = "TIME i p r i ": 4 2 cu 1 7 r 3 where a W = "TIME i x "i 5 cu

= 1 I 5 1

W 5

75

76

CHAPTER 2

1 fi a r = 1 4 7

I 1

1

1

W

W 6

1 I false 6 |

1] 10

9

false

11 ii false

7 false

V 6

W 7

where

« = "TIME i T + t ",

W 6 W 7

= TV ; 6

] I S I1 8 r = I 5 1 I 12 i i 12 1 I 14 I

1 12

1

W 8

W 9

W

W a a

9 9 W 8

13

W 9

where W

= "c(l 8

a i i , TIME] = 0" V "x i x "; 12 cu cu

i where x is the current characteristic of the a -token, cu i

which is lo-

cated in place 1 ; 12 W 9

: iV . 8 Tokens

a , a , . . ., a 1 2 t

with initial characteristics: "T " eni

ter 1 at time-moment TINE = T and after this they enter place 1 1 3

im-

mediately. There they receive new characteristic "j)(T , TIHE), i

KC , Q(T , C , TIME), P(T , C , TIME), L(T , C , TIHE) > j i j i j i j / C € j

<MT , TIHE] J, i

wtoere p(T , TIME) is the vertex V i k

where the TS

T

is located. i

Si-

SCME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

multaneously with the above transfers,

the token

7T

p enters place 1 2

with an initial characteristic "(V , M(V , TIME!), [ k k k k' k k' R k' / V € NDN(V ))", kJ k where NDN(V ) is the set of all next-door k

neighbour vertices

of

the

vertex V . The a-tokens split in places 1 (if the TS T has some posk . 3 i sibilities for motion) and 1 (if the path for motion 4 unique).

of the TS T is i

In place 1 some tokens mist be generated which have one li4

near predecessor.

All tokens which have equal initial characteristics

"T " are different representatives of the i

token

a i

and there

is no

concurrent situation among them Note that concurrent processes do not exist in this GN model, but such processes exist in other GN-processes of the TS-problem. The a-tokens do not receive any new characteristics in place 1 , but in place 1 they receive the characteristic 3 4"V , TIME + D(V , V )", k' k k' k' k k' where V kJ

is the vertex to which the TS T will come. After the transi

fer from 1 to 1 , 4 5

token a receives the characteristic

a a "TIME + N(pr x , pr X , TIME)" 10 1 cu and in place 1 - the characteristic 7 a a a a "R(pr x , pr x , TIME) - E(pr x , pr x , TIME)", 1 0 1 cu 1 cu-2 1 cu-1 which is the net profit for the vertex. When the a -token enters place i

78

CHAPTER 2

1 , it receives as current characteristic the name of the vertex, whe9 re it is located at these

tokens

the current moment, i.e. p(T , TIME). In place 1 i fl

receive

as a characteristic

the value

of the

tokens

transfer in the net according to the criterion HG. Finally, in the frames of a transition Z 5

the a -token with the i

maximm value for HG is determined, The token

p

is transfered in the

transfers of the a-tokens. current characteristic

GH

simultaneously with the

It enters place 1 , where it receives as a 6

the new values of the parameters

which it has

in its previous characteristic. Thus the process of the transfer of

TSs is described.

For the

J

GH s definition the following data are also necessary: (a) the tokens transfer in a package, i.e.

in the frames

of one tact

all tokens which can be transfered from input to output transition places, do this. o
is the minimum time which

is related

to TS-stays or TS-transfers (in real situations, this can be e.g., 1 min. ). (c) the transitions are activated in every time-moment about the fixed 0

time scale with parameters T , t i

and

• t = T - T. i f

o (d) the duration of the transition's activities is t . Therefore o e (t) = t 4 t . 2 (e) the tokens transfer is determined by their time-characteristics. (f) the capacities of all arcs and places are oo, without

SC*E APPLICATIONS OF GENERALIZED NETS IN SCIENCE

79

C(l > = c(l ) = C(l ) = c(l ) = 1. 2 6 10 11 (g) the priorities of all places

and transitions

are equal,

without

Places 1 , 1 and 1 for which it is valid: 0 14 12 n (1 ) = n (1 ) > Tt (1 ). L 6 L 14 L 12 (h) o 1

= A(V(1 , 1 , 1 ), v(l , 1 >), 1 3 9 2 11

(i) D = A(1 , 1 ). 4 6 7 The other transitions are of a disjunctive type. Thus a GN in frames of which the ETTSP1

is solved

is construc­

ted. This GN has some faults and the reason for them is

that the num­

ber of oc-tokens can grow un-1 .unitedly. Below we problems and

shall introduce some other modifications

of the TS-

discuss the possibilities of their descriptions with the

GNs: (ETSP2) Trace out the locality-optimal (for a criterion LMG: ty-maximal gain") TSs T , T , . . . , T 1 2 t

route of the i-th TS

(l <, i i t> which mist move in the built-

up areas V , V . . . , V 1 2 v .. . , C c

when

"locali­

among the independent

carrying some of the products

the different TSs

do not Know

C , C, 1 2

the plans of the

others. (ETSP3) Trace out the locality-optimal (for a criterion LMG): route of a in number groups of TSs:

(T ,T 1,1 1,2

T

), l,t

(T 2,1

1 T

, . . . , T 2, 2 2, t 2

),...,

move in the built-up areas

IT ,T a, 1 a, 2

T ) Which a, t a

V , V , . .. , V , 1 2 v

must

carrying some of

80

CHAPTER 2

the products

C , C ,..., C ; 1 2 c

the TSs

from d i f f e r e n t s e t s do

not concur. The last problem can also be defined for deterministic

and for

non-deterministic cases. Variants of the above problems are those where

the TS

into all the V-places or go to some of these places only, TS must

have exactly determined qualities

must go

where every

of products or where every

TS must travel certain exactly determined distance. The above TS problems can also be described by QHs as above. This paper is based on [8,9]•

REFERENCES: [1] E. Lawler,

J. Lenstra,

A. Rinnooy Kan and D. Sbmoys

(Eds.), The

traveling salesman problem. A guided tour of combinatorial optimi­ zation, H. Y. : Wiley X Sons, 1985. [2] I. Helamed, S. Sergeev and I. Sigal,

The traveling salesman prob­

lem Issues in theory. Avtomatika i telemecbaniKa,

1989,

9, 3-33

(in Russian). [3] I. Helamed, S. Sergeev and I. Sigal,

The traveling salesman prob­

lem Exact methods. Avtomatika i telemechanika, 1989, 10, 3-29 (in Russian). [4] I. Helamed, S. Sergeev and I. Sigal,

The traveling salesman prob­

lem Approximate algorithms. Avtomatika i telemechanika, 1989, 11, 3-26 (in Russian). [5] J. Little, K. Hurty, D. Sweeney and C. Karel, An algorithm for the traveling salesman problem

Operations Research,

Vol. 11,

1963,

972-989. [6] T. Volgenant and R. Jonker, On some generalizations of the travel­ ling salesman problem,

Journal of Operational

Vol. 38, Ho. 11, 1987, 1073-1079.

Research

Society,

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

[7] R. Ackoff

and M. Sasieni,

Fundamentals of operations

81

research.

John Wiley and Sons, Inc. , New York, 1968. [8] K. Atanassov,

A generalized net, representing

the travelling sa­

lesman problem. AMSE Review Vol. 14, No. 4, 1990, 61-64. [9] K. Atanassov, Generalized net and extensions of the travelling sa­ lesman problem press).

AMSE

Review

Vol. 21,

No. 2,

1992,

19-26

(in

82

CHAPTER 2

§ 2. 3: GENERALIZED NETS AND LABYRINTHS Erassimir Atanassov and Rumen Cbristov

Different researches on the methods for exist (see e.g., [1-3]).

motion

in a labyrinth

Here we shall describe one of them, which is

based on the GNs. Let the labyrinth L be given. It can have the form of

(m x n) -

dimensional (0, 1)-matrix, the elements "0" and "1" of which correspond to the labyrinth corridors and walls, respectively. We construct

the GN from Fig. 2. 5. It has two transitions with

transition conditions

l1

3 3

m I false II m |false 2 1 r = I 1 1 I W 1 1 3 1 2 1 55

where

|I 1 I ||

W 3 W 33

l1

l1

4 4

false false

5 5

mm

false false

m m

2 2

3

W

W 1 2 W 1 2

W

w 4

W w 5

false

false

w 4 W 44

w W 5 W 55

false

false

false

false

W = "c(l , TIME] + c ( l , TIHE] > 0" 8 " c ( l , TIHE) = 0"; 1 3 5 3 W

= nW ;

2 W

1 = "end of the (unique) path or a meeting with another

token on the

4 path or along this path another token has already passed"; - IV ;

W

3 W

4 = "there is more than one path

5

and along this path

has not passed";

and

1 66 r

1 2

1

22

| W W W W 3 | 6 7

another token

SCME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

83

where W

= "the token arrives to an exit"; 6

w = -m .

7

Fig. 2. 5 First,

toKens a and p enter the net through places 1 and m , 1

respectively and

p has an initial characteristic in the form

1 of the

above mentioned matrix. If there exists exactly one path for motion, it goes to place 1 with a characteristic "nurber of this path, nurabe3 red

according to the hour hand".

If the number of the paths

is more

84

CHAPTER 2

than one, the token a splits sequentially to the tokens a , a , . . 1 2

(s is the number

time one token enters place 1 3

with the above characteristic

one (which will generate the next tokens) enters place 1

s

Each

of the different paths in front of token a ) .

another

without cha-

5

racteristic.

when there are no paths for motion

a igiven

in front of

token, it goes into place 1 with the characteristic "unsuccessful end 4 of .a motion". place 1 6

If there is an exit in front of

with the characteristic

a token,

it goes into else

"successful end of a motion";

the token enters place 1 without a characteristic. 2

All 1-places have equal priorities,

which are bigger

than the

priorities (equal) of all m places. Both transitions have equal rities. All places have equal (oo) capacities,

prio-

the temporal components

are standard and thus not given. The problem of aspects.

motion in

a labyrinth can

be extended in some

We shall show one extension and its GH-representation:

the labyrinth has some floors GH, we shall

(i.e. it is not planar),

when

in the ;above

change only the characteristic function of place 1 3

"number of the path, numbered as first underneath - uphill and

with

as se-

cond along the hour hand" and the initial characteristic of token P is (m :K n x k) -dimensional (0, 1) -matrix. REFERENCES:

tl] F. George, The foundations of cybernetics,

Gordon and Breach Sci.

Publishers, London, 1977.

[2] H, Gardner,

Mathematical puzzles

and diversions,

Bell and ,Sons,

London, 1965.

[3] K. Shannon,

Research on

theories of informatics and cybernetics,

Inostrannoi Literatury, Moscow, 1963 (in Russian).

SCHB APPLICATIONS OF GENERALIZED HETS IN SCIENCE

85

5 2. 4: FORMAL COMMUNICATION IN SCIENCE: A MODEL BASED ON GENERALIZED HETS Fadosvet Todorov

§ 2. 4. 1

and

Krassimir Atanassov

Introduction

Various studies

of the

been carried out (see [1-5]). plain the informal and/or

connunication network in science have Different models have been used

formal connunication

processes.

to ex­

Models of

forma] connunication in science are predominantly linear and emphasize various media (articles, journals, books, etc.), participants (indivi­ duals and institutions) offers two views

or functions

(activities).

In [1], Atherton

of the connunication network of scientific and tech­

nical information, one of which stresses for example

the place of the

publisher. Garvey presents (see [2]) a schematic overview of

the sci­

entific communication system which includes all events that occur from the time nal)

the scientist begins his research,

publishes it (in a jour­

until its appearance in reviews, treatises, etc. Also Garvey de­

scribes in [3] features of the current dissemination system in science and suggests some innovations in it. Some linear models of the formal connunication process are pre­ sented in a recent review article by Hills ([4]). The review "the scholarly

connunication process in terms

looks at

of the interaction and

concerns of the scholar, the learned society, the publisher, the pro­ duct, the librarian, and the new technology". King

et al. offer (see [5]) a schematic overview of the disse­

mination of scientific and technical research results by different me­ ans and of the general flow of information among scientists and neers.

The authors also supply a stochastic model of the

engi­

transfer of

article manuscripts between authors and publishers in order to descri­ be the system of journal publishing in terms of cost and flow of mate­ rials.

86

CHAPTER 2

Here we shall present at outline of a formal communication pro­ cess by stressing

some

for the traditional

specific functions and participants required

transfer of

(originator) to the reader

article manuscripts

from the author

(consumer). The purpose of the study is to

show a possible description of this process by means of a mathematical model with GHs.

§ 2. 4. 2 Schematic overview o± formal comami cation Here we choose

a simplified

part of

in

science

the formal communication

networK in order to apply in a comprehensible manner the GH (having in mind the

unacquainted reader).

under consideration,

The schematic overview of the process

namely the publishing of article manuscripts and

distribution of journal articles, preprints in Fig. 2.6,

or reprints, is presented

The author (A) submits an article manuscript to the edi­

tor (s) of a given journal. "Hie flow of manuscripts between authors and editors is represented by arrow 1 reject obvious

a paper whose subject matter is inappropriate

editor then

This is

symbolized by arrow 2.

transmits the remaining manusrcipts

they believe to be experts this

or which is "an

non-starter" [6]. A recommendation may be made to submit the

manuscript elsewhere. the

(see Fig. 2.6). The editor (E) may

in the particular

activity is represented by arrow 3.

(F) returns the evaluated paper (arrow 4) commendations for acceptance not need re-examination),

"to people whom

topics involved"

The expert or

[6],

the referee

to the editor with his re­

(without change),

improvements,

Traditionally,

corrections

modifications or

(that do

rejection.

Only in the first case (direct acceptance) is the manuscript transfer­ red to the

publisher

(arrow 5),

who in our diagram is to be related

with recording and reproducing activities rather than being corporated in the publishing process. If the manuscript is rejected or a revision is recommended, then the author is informed (arrow 2). The author then

SCME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

87

decides whether to revise the manuscript, to send it to another jour­ nal

(usually

without

modifications)

or

to give up

publication of

(withdraw) his paper. After manuscript recording, the publisher sends proof-copies to the author (arrow 6).

The corrected proofs are returned for reproduc­

tion and issue preparation (arrow 7). The published articles reach the user (U) through the

distributor (D), who incorporates the functions

of such participants in the formal communication process as publisher, librarian, information services, etc. Fig. 2. 6

also shows

the direct contact between the author and

the user. The significance of arrow 9 is a reprint request and that of arrow 10 correspondingly the satisfaction of the request.

Fig. 2. 6 Schematic overview of the formal conmunication by means of journal articles (A: Author, D: Di­ stributor, E: Editor, P: Publisher, R: Referee, U: User)

88

CHAPTER 2

§2.4.3

A GS model Fig. 2.7 presents the graphical structure of

a GH which models

toe relations discussed above (see also Fig. 2.6). The circled capital letters are the places Where different activities are performed ting, reviewing, etc.). f

are "fictitious"

(edi­

places which are introdu-

i ced

to conform to

the graphical

representation of the GH.

They are

used to determine the length of time needed hy the tokens to pass from a "real" to a "real" place. The f are "fictitious" because the toKens 1 do not receive new characteristics as they transit through the f . i The token (author)

(article manuscripts)

enter the net through place A

with the following initial characteristic "name,

the author,

manuscript's title".

Place A

address of

is the entry point for all

tokens which represent the manuscripts ready for a submission to a jo­ urnal. Transition condition r

has the trivial form 1 l ff 88

ff II true true 7 II rr

=

R R

II true true II A I I true true A

1

ff II true true 4 II Transition condition r

has the form 2

E r

= 2

f

| W 8 | 1 I R I true

f f 2

f f 33

W W

ff

W 2

W 3

W 4

W 5

false

false

false

false

4

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

89

Fig. 2. 7 where z "A takes the decision to submit a manuscript";

W i W

= "A is ready to return the proof-copies to P"; 2

W

= "A is willing to send a reprint to U"; 3

W = "A decides not to publish or to withdraw his manuscript" 4 (The place W is an output place of the model. This implies that A will take no further steps to publish the paper under conside­ ration);

90

W

CHAPTER 2

= "A agrees to make changes in order

to publish

the manuscript in

5 the same journal or decides to resufamit it without

improvements

to another journal". When the GH's token

(which models some manuscripts] arrives in

place E, it receives the characteristic "journal title". Transition condition r

r

:

E

has the form 3 R f i 6 1

I W 1 6

3

W 7

W 8

where W

= "The editor E takes the decision to inform the author A" 6 (The notice

could include

a rejection of

the submitted manu­

script, a reconmendation for revision, etc. In other words, the token arriving in place A

(coming from E)

receives the charac­

teristic "information from E to A"); - "E decides to send the manuscript to be assessed by an expert R";

W 7 W

e

= "E takes the decision to transfer the manuscript to the publisher F in order to be published (after peer reviewing]". Transition conditions r

and r 4 U

f

--

r 1

I true 3| _ I D I true f

I true 5 I

and P r

=

f

5 f

I true 1I

I true 2 I

have the trivial forms 5

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

91

Transition condition r has the form 6 b

r = e e

P U

f 7

D

| i i

W

W

I I I

W

f 5

9 9

to 10 W

12 12

W it li W

13 13

14 14

where W

= "P is ready to send manuscript copies to A" 9 (These are the proof-copies or the reprints according

to the mo­

del time), = "P is ready to send the issue containing the article in question

W lO

to D", z "P is ready to send the Journal issue directly to U",

W 11 W

= "U wants to send a letter

(with a comment)

or a request for a

12 reprint to A", W

s "U is willing to receive information or the original from D", 13

W

= "U

is willing to be disseminator of

the article

for

another

14 user". Entering f

from P, the token has the characteristics

"proof-

7 copies" or "reprints". Arriving in D from P, it receives the characte­ ristic "bibliographic data". When the token enters f from U (through 6 f ) it has the characteristic "Kind of information or requested mate7 rial". K

Let the GN function between the tune -moments T and T + t . Ini­ tially (at the moment T) the tokens enter the net in ve for E. The predicate W

place A and lea­

has the current value "true" and the remal1 ning predicates in the transition condition r have zero-values. From 2

9E

CHAPTER 2

place E

the tokens move to places A or F according to the editor's

judgement. Visiting places A and E the toKens receive the above menti­ oned characteristics. When a token returns to f , it could enter place f (modifica8 4 tion of the manuscript), W (withdraw) or f (sending a reprint). If 3 the token enters R coining from E it returns obligatorily to E, and ac­ cording to the characteristic

received in R (referee opinion) arrives

at A or P. From P it moves to A with the characteristic "proof-copies" or "reprints". The token with the characteristic "proof-copy" goes from f to P through f . With a new characteristic (bibliographic da8 2 ta) the token arrives in D or f coming from P. The ultimate aim of 5 the tokens is to get to X). notice that in our case, the place f sym8 bolizes the author's activity and U that of the reader, i.e. we do not consider

(for the sake of simplicity]

the possible manifestation of

the user as an author and correspondingly that of the author as a rea­ der of publications. § 2.4. 4

Discussion

The proposed diagram of formal conmunication 2.6)

does not show all possible

the process. as well,

in science

(Fig.

interactions among participants

in

In the scheme, there could be represented other contacts

such as a possible

direct link between the author and

the

referee: the author could send an accompanying note to the referee and the latter could openly publish (or answer individually) his recomendations for manuscript Improvement. All the simplifications as well as some artificial divisions are made here to perceive more easily the GH as a modelling tool appropriate

to conmunication network.

The model

based on the GHs is sufficiently powerful to be used for the descrip-

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

93

tion of the whole (ccnplex) structure and dynamic picture of knowledge generation, representation,

distribution and use, or for detailed re­

presentation

part of

of a specific

minimum modifications

the conminication process.

With

(simplifications), the model in Fig. 2. 7 can be

used for example, for the description of booK publishing and distribu­ tion. Fig. 2. 6

can be also presented

by means

of

a block diagram,

which would reflect the interactions among participants lopment of the

and the deve­

process under consideration. But a block diagram

can­

not describe the simultaneous movement even of two or more manuscripts submitted at the same time to one, two or more journals. neous

transfer of two (or more) manuscripts in the network can be de­

scribed by means of ons,

The simulta­

e. g. , the Petri nets and their other modificati­

but with the increase

in the number of manuscripts, it would be

impossible to follow the individual fate of the manuscripts at various places of the net. The transfer of a given number sented by

of manuscripts

can

be

repre­

the stochastic model of King [5]. However, in this case one

cannot follow

the dynamic development of the process or "predict with

certainty the time

required to perform any of the article preparation

functions, nor whether the articles will be rejected, accepted, or mo­ dified" [5]. Only by means of GNs it is possible to follow manuscript (on the basis of the different ved by the

the history of a

GN's characteristics recei­

tokens at each GN's place) and to trace its propagation in

a time-frameworks.

§ 2. 4. 5 Possible

applications

of the GNs

The simulation of a formal comninication process will on

the

values of the transition condition predicates,

be based

which can

be

94

CHAPTER 2

determined by the token's characteristics or on the basis

of experts'

assessment. Using the results of the program realization and simulati­ on,

a wide variety of questions can be answered and various conclusi­

ons can be drawn. The quantitative results may ticles

include: published ar­

(on a certain topic or in a given Journal)

all the submitted manuscripts; without revision]

as a percentage of

or directly accepted manuscripts (i.e.

as a percentage of all

the accepted article manu­

scripts; the number of resubmitted manuscripts that are accepted; number

of

submitted

etc. The qualitative how

does

the number

manuscripts by authors with

of

submitted manuscripts

change; is there any relation characteristics

the same

answers are to questions such as on

is the flow of

a specific topic and initial

submitted manu­

scripts

sufficient for the "safe"

Through

this model many time-related questions can also be

One could

appreciate

address,

the following:

between the rejection rate

of the manuscript;

the

existence of a given journal, etc. answered.

the mean model time needed for the transfer

the manuscripts from place E to place D

of

(publication time-lag) or the

time required to pass a particular place, etc. This paper is based on [7],

REFERENCES: [1] P. Atherton,

Views of the communication network of scientific and

technical information. International Forum on Information

and Do­

cumentation, Vol. 1 (1975), Ho. 1, 10. [2] W. Garvey et al. , Research studies in patterns of scientific com­ munication, Inf. Stor. Retr. Vol. 8 (1972), 111-122, 159-169, 207221, 265-276. [3] W. Garvey, S.

Gottfredson, banging the system: Innovations in the

interactive social system of scientific conmunlcation. Information Processing and Management, Vol. 12 (1976), Bo. 3, 165-176. [4] P. Hills,

The scholarly conmuni cation process.

Annual Revies for

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

Information Science

95

and Technology, Vol. 18 (1903), 99.

[5] D. King, D. McDonald, N. Roderer, The journal in scientific conminication:

The roles of authors, publishers, libraries and readers

in a vital

systen.

National

Science Foundation Contract NSF-C-

DS175-0694-2. RocKville, Mary lands, King Research, Inc. , May 1979. [6] A. Meadows,

The problem of refereeing. The Scientific Journal (A.

Meadows, Ed. ), Aslib Reader Series, Vol. 12 (1979), London, ASLIB, 104-111. [7] Todorov R. , Atanassov K. Formal communication in Science: based on Generalized nets, 177-185.

A model

Scientometrics, Vol.9 (1986), No. 3-4,

96

CHAPTER H

§ 2. 5: GENERALIZED HETS AND EXPERT SYSTEMS. V Krassimir Atanassov,

Peter Georgiev

The main problem in designing, Systems (ESs) is to select lism (see e.g. [1,2]). i.e.

and Martin Tetev

Implementing and

using

Expert

the proper Knowledge representation forma­

Host of the ESs apply the rule-based approach,

they represent the knowledge in "IF situation THEN action" man­

ner. The papers [3-6] deal with another means of representation, main­ ly with a generalization of Petri nets

and discuss how the rule-based

approach can be viewed through the new formalism with all the ensuring consequences.

On the other hand the GHs are an

extension of the well

Known Petri nets allowing a detailed description of parallel

(or con­

current) processes. The GN described in the paper can be considered as an extension of the nets from

[3-6).

All notations of [3-6] are saved here.

This

gives the possibility to trace the development of the idea for the GHmodelling of the ESs. On

the other hand

we introduce new idea related to

the ESs,

based on the GH-description possibilities. The main components of a production (DB)

containing

system are:

the Data Base

the facts about the problem which are

to be solved,

the Knowledge Base (KB)

containing the rules which are

to be used in

the reasoning process, and the inference engine which operates through the KB using the DB for proving or rejecting rules in form,

the KB

are supposed

to be just

a given hypothesis. in a positive

The

conjunctive

since otherwise the rule can be split into a number of

"subru-

les" of that type. We shall describe the ES J s facts with ties.

How we can compare

priorities.

new components - priori­

two arbitrary facts on the basis

This possibility is very

useful

of

their

in the particular cases

when both facts coincide or when both facts have contrary senses.

SCHE APPLICATIONS OF GENERALIZED NETS IN SCIENCE

97

Let to every fact A of KB he Juxtaposed by a natural nuntoer u A which corresponds to tbe priority of A. Let tbe new fact B with a pri­ ority \i be generated by some way in a certain time-moment when tbe B ES functions.

If both

facts

are not related,

ters the DB. In the ordinary ESs, fact

A, when

function

the new fact

then the new fact en­ B substitutes the old

B coincides with, or contradicts to A. Now the

in another way, basing on tbe new component,

ES will

when tbe facts

A and B coincide, tbeir representative (in particularly - A or B) stay in tbe DB, but with a new priority - tbe maximm of \i and \> . On tbe A B

other band, tbe fact with tbe mcnrimm priority between \i and \> stays A

B

in tbe DB when tbe facts A and B are in a contradiction. The constructed here GN (see Fig. 2. 8) represents the ESs with data's priorities in tbe above sense. There have been several attempts to describe production systems by some Kinds of nets (see cf. [7-15]). Following [3-6] some definiti­ ons

will be given here in order to describe

different components

of

tbe GN appearing below. Let a token

a enters place 1 of the GN with initial cbaracte1 a a ristic x {- "a hypothesis") and let z denotes its current cbaracO last teristic. Let a token

b b enters place 1 with initial characteristic x (= 2 O

"DB"). Let tbe token

c enters tbe place 1 with initial cbaracterisric 3

c x

(= "R"), where 8 : IF , I 1 I is tbe list of tbe rules. Each 0 1 2 n rule E has the form , where C is tbe consequi i 1, 1 i, s i i

98

CHAPTER 2

ent and A

A 1,1

are Uhe elements of the conjunction

presen-

i, s i

ting the antecedent,

z

denotes the i-th characteristic of

the token

i with the highest priority in a given place. Finally, let toKens d , d , ..., d 1 E s

(s z 0) enter place 1 wuth 40

d cu initial characteristics x

(= <"new fact", "its priority")), where

0 "cu" symbolises the current number of the d-type token, Z

The transitions in the GU are the following. = 9 40 2 33 48 41 42 46 9 9

where 1 41 41 1 I 40 I r

= 9

W 30

1

1 42 42 W

1 43 43 W

44 44

1

1 45 45 W 34

false

46 46

31

32

W 33

1 | false 2 I

false

false

false

false

true

1 I false 33 I

false

false

false

false

true

1 I false 48 I

false

false

false

false

true

where d d d cu cu p cu cu cu W W = "pr "pr xx ee xx " S "pr "pr xx 22 Q(pr Q(pr xx )", )", 30 30 1 00 llast ast 2 0 10 where Q(A] is the priority of fact A, if A € x

B lasr

P « x ; last d d d cu cu p cu cu cu W :: "pr W "pr xx €€ xx " 88 "pr "pr xx < Q(pr Q(pr xx J"; J"; 31 31 1 00 1 llast ast 2 00 2 11 0

and Q(A) = 0 ,

if A

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

d cu W 32

z "ipr : "lpr x l o

d cu

p ex last

" & "pr x 2 0

99

d cu i Q(npr x )", 10

where TA is the negation of the fact A; d cu W 33

- "ipr x l 1 0

ex last

d cu w

and D 9 Z 10

" s & "pr x 2 0

d cu < Q(npr x )", 10

d cu p " & & "npr x i x )" last 2 0 last

p i x

: "pr x 34 1 0

d cu

p

= v(l , 1 , M l i , v(l , 1 , 1 )); 2 33 40 2 46 33

= <(1 J 1, , fl (1 , 1 ), K, *, r , K, v(l )> 46 47 46 lO 46

where

r 10

1 47 47

1 48 48

: 1 I W 46 I 35

W 36

where W

= "c(l 35

W

, T H E ) = 0"; 40

r "C<1 36

, TIME) > 0"

40

where c(l, TIME) is the number of the toKens in place 1 at the current time-moment TBE; Z : <(1 , 1 , 1 , 1 1 where

1

47

3

35

!, fl , 1 4

5

1 !, *, *, r , K, D > 9

(here we shall use the notation from

ding corrections)

1

1

[3-6] with the correspon­

100

CHAPTER 2

1 4 1 1 r

= 1

1

1

1

5

I W 1 1

1

6

w 2

false

false

false

W

false

false

false

1

I false 1

false

false

false

W v W W vW 1 2

1 || ffalse alse 35 I 35 I

ffalse alse

ffalse alse

ffalse alse

W W vv W W W W vv W W 1 2 1 2

where the predicates W, W

and W 1

v W 1

9

false

1 I false 47 I 3

W

1 8

7

v W 2

have the form 2

W = "there are more tokens before place 1 "; 1 a b W W = x € x ;; 2 00 0 W W 1

= = nW ;; 2

and DD 1

= A(I A ( I ,, ll ,, v v(l ( l ,, 1 >); >); 11 47 47 33 35 K

Z ss 2 4 6 8 10 12 13 14 15 16 2 2 2 4 6 8 10 12 13 14 15 16 2 2 where 11

r

= 2

K K

lO lO W 5

11

12 12 W 6

11

13 13 false

11

11 14 14 false

11 15 15 false

16 16 false

false

false

1 4

I 1

1 6

| false 1

false

W v W 5

W v W 6

1 8

| false 1

false

false

false

where the p r e d i c a t e s W and W have the form 5 6 c a W = "mnt>(pr x , x ) i- 0"; 5 1 0 last c a W = "mnf>4pr x , x ) - 0" 6 1 0 last

W v W W vW 5 6

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

where nun*>(Y, y) the ordered set

101

is the number of appearances of the element y in the Y.

(In [3-6] there are other predicates related with K

places 1 , 1 and 1 4 10 12

Here place

the GN from [6]. This subnet

1 10

is an input to the subnet of

realizes the process of checking of the

hypothesis and it is aside of the present model); Z = 6 13 31 29 31 6 13 where 1 29 29 rr 6

= 11 II W W 13 13 II 16 16 1 II W W 31 II 31 16 16

where the predicates W

and W 16

W

1 31 W W 17 17 W W 17

have the form 17

= "there are no tokens in the subnet" 16

(the form of this predicate in [6] is similar to the above); W

= "a new fact is generated in the subnet" & "there are some tokens 17 in the subnet";

Z = <(1 , 1 !, (1 , 1 , 1 , 1 i, *, «, r , *, D > 7 29 15 32 33 34 35 7 7 where

rr 7 7

l1 32 32

33 33

= 11 ||W W vvW W 29 II 18 18 29

W W vvW W 19 19

1 30 I 30 I where the predicates W

and W 1«

W

= "the 18

W = nW . 19 ia

ffalse alse

ii

ll

ll 35

3434-

ffalse alse

ffalse alse W W vv W W 18 18

false false W W vv W W 19 19

have the form 19

GN-modelled ES is not a self-addapted system";

102

CHAPTER 2

Fig. 2. 8

SC*E APPLICATIONS OF GENERALIZED NETS IN SCIENCE

103

The characteristic functions f of the tokens associated to the i corresponding place 1 are as follows: i $ 5 $

a a — > "!x " (the hypothesis x is valid); 0 0

a a — > ""Mix )" (the hypothesis x is not valid); 12 o 0

»

$ 31

— > "x U fx), where last which

(The place

x is the last characteristic of the token

has entered place 1 " 20

1 20

is an element of the subnet.

In it enter tokens with

current characteristic "new fact", which is received in the process of a checking of the given hypothesis x ); 0 o d d d / p cu cu cu / (x - (<pr x , Q(pr x )>) U {x i 0 last 10 10 , if the token d is in place 1 cu 41 I •* I

$

46

— —>>

(

\ I

, if the token d is in places 1 or 1 CU 42 44 d

d d cu cu cu (x - (<pr x , Q(npr x )>! V (x ) last 10 10 0 p

, if the token d is in place 1 cu 43 p

cu

x U (x V last 0

1 , if the token d is in place 1 cu 45

(the other functions are not defined). The GN considered in [3-6] ting production systems

shows the possibility

of represen­

(their functioning and their results) by GNs,

This possibility follows also from the fact that a GN can describe the functioning of systems were

Predicate/Transition nets,

and in

[7-15]

production

represented by means of Petri nets, Predicate/Transition

10*

CHAPTER 2

nets and other Petrl net modifications. [3-6]

have the following

The GNs described here and in

universal property:

it is not dependent on

the particular modelled production system These GNs are not connected with the description of

the parti­

cular ES. It should be mentioned that existing

(Known from the literatu­

re) nets representing production systems are the particular system they describe and

rigidly

associated with

cannot be changed (deformed).

On the other hand the two GNs from [121, the second GN from [3, 4] and the GNs from

[5, 6]

they represent.

are maximally

independent of the particular

ES

Finally, the described here GN includes as particular

cases all previous constructed GNs describing ESs. REFERENCES: [1] Building

expert

systems,

F. Hayes,

D. Waterman

and

D. Lenat

(Eds. ) Addi son-Wesley Publ. Co. , Reading, Mass. , 1963. [2] D. Waterman, A guide to expert systems, Addison-Wesley Publ. Co. , Reading, Mass. , 1906. [3] K. Atanassov, L. Atanassova, E. Dimitrov, G. Gargov, ski, M

Marinov, and S. Petkov, Generalized nets

tems. I. 12-th

Methods of Operations Research,

Symposium on

Mathematik,

I. Kazalar-

and expert sys­

Vol. 59.

Proc. of the

Operations Research of the Gesellschaft

OKonomie

und

Operations

Research,

1987,

fur

Passau,

Frankfurt a.M: Athenaeum, 1989, 301-310. [4] K. Atanassov, L. Atanassova, E. Dimitrov, G. Gargov, ski, M

Marinov, and S. Petkov, Generalized nets

tems. II. IFIP Symp.

"Network

Information

I. Kazalar-

and expert sys­

Processing Systems",

Sofia, May 1988, Vol. 2, 54-67. [5] K. Atanassov, L. Atanassova, E. Dimitrov, G. Gargov, ski, M

Marinov, S. Petkov and M

nets and

expert systems. III.

Stefanova-Pavlova,

Methods of

Operations

I. KazalarGeneralized Research,

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

Vol. 63.

Proc. of the

14-th

105

Synposiim on Operations

Research,

Ulm, Sept. 1989, 417-423. [6] K. Atanassov, L. Atanassova, E. Dimitrov, G. Gargov, ski, M

Marinov, and S. Petkov, Generalized nets

tems. IV.

Proc. of the XIX

Spring Conf. of

I. Kazalar-

and expert sys­

the Union of Bulg.

Math. , Sunny Beach, April 1990, 155-161. [7] M. Bonacina, Petri nets for Knowledge representation,

Petri Nets

Newsletter 27, 1987, 28-36. [8] J. Duggan and J. Browne, ESPNET: Expert-system-based simulator of Petri nets, IEE Proc. D

(Control Theory and Applications),

Vol.

135, No. 4, 1988, 239-247. [9] A. Giordana and L. Saitta,

Modelling production

rules by

means

of predicate/transition networks, Inf. Sciences 35, 1965, 1-41. [10] U. Mainz, Netztheoretische reprasentation pradikatenlogischer begriffe und methoden, Diplomarbeit, Univ. Bonn, 1985. [11] A. Sinachopoulos,

Derivation of

a contradiction

by

resolution

using Petri nets, Petri Nets Newsletter 26, 1987, 16-29. [12] S. Stoeva

and

K. Atanassov,

Generalized net

production systems interpreters,

Proc. of the

representation of Fifteenth

Spring

Conf. of the Union of Bulg. Math. , Sunny Beach, 1986, 456-464. [13] R. Valette,

Nets in production

systems,

Proc.

of an Advanced

Course "Petri nets: Applications and Relationships to dels

of Concurrency",

Bad Honnef,

Other Mo­

in Lecture Notes in Computer

Science, 255, 1986, 191-217. [14] K. Voss, Nets in Data Bases, Proc. of an nets:

Applications and Relationships to

rency" , Bad Honnef,

in

Advanced Course

"Petri

Other Models of Concur­

Lecture Notes in Computer Science, 255,

1986, 97-134. [15] D. Koutanis, R. Rasshidi, expert systems. 1987, 143-152.

Petri net representation of rule based

First Annual BSD/SMI

Expert Systems Conference,

106

CHAPTER 2

§ 2.6: GENERALIZED NETS RESEARCHING THE BASIC PROPERTIES GP KNOWLEDGE BASE Ewgeni Dimitrov,

U y a KazalarsKy

and Krasslmir Atanassov

One of the basic elements of every

Expert System

(ES)

is the

Data Base (KB). The structure and correctness of the DB determine the properties of the corresponding ES. The Knowledge in the ES

is repre­

sented as Production Rules (FRs). The possibility

to check the correctness of

the KB

is a very

important condition for the functioning of the ES. We shall construct a reduced the existence

GN

of at least two rules in

(see App. 5) which determines a given EB,

having identical

antecedents (see Fig. 2.9). The capacities of its places and arcs are

Fig. 2.9 infinite. I t s transitions are as follows: Z = 1 1 2 3 4 1

SOJE APPLICATIONS OF GENERALIZED NETS IN SCIENCE

107

where 1l 3 1 II W W r = 11 II ll r 1 || 1 II W W 2 11 11

1 4 W W 2 W W 2

where W = "the place 1 i s empty" or "in place 1 there e x i s t s at least one 1

3

3

token with the characteristic - the same

antecedent

as in

the

present token",

w = -m ; 2

1 Z 2

= <(1 1, (1 , 1 ), r > 3 5 6 2

where 1 5 5 r 2 2

= 1 I W 3 1 3 3 1 3

1 6 6 W 4 4

where W

= "the transitions Z 3

and Z are not active" and "at the moment of 1 3 activation of the transition Z in place 1 there exists only 2 3 one token"

(this token enters place 1

and leaves the net;

if it is

the unique

5 token of the net, it receives as

a final characteristic

"there exist

no two PRs with identical antecedents"); W = "the transitions Z and Z are not active" and "at the moment of 4 1 3 activation of the transition Z in place 1 there exist at le2

3

ast two tokens" (these tokens enter place 1 6

and leave the net with a final characte-

CHAPTER 2

loa

ristic "the PRs which are Initial characteristics of these tokens have identical antecedents"); Z 3

= <(l !, (1 ), r > 4 2 3

where

1 2 r

: 1 3

I W 4 | 5

where W = 5

'the places 1 and 1 are empty". 1 2 All PRs of a given KB can be described as initial

characteris-

tics of the tokens of the above GN and thus an answer for tic question of the existence of identical antecedents in PRs can be obtained. The GN

which checks the set of al1

PRs

of a given

the existence of at least two rules frcni the form has the form as in Fig 2. 10.

KB

about

A I- C and A & B I- c

The transitions Z and 1

Z 3

in both nets

coincide with those from Fig. 2. 9 without

w - • the 1

place 1 is empty" or "in place 1 there exists at least one 3 3

token with the characteristic - the same

consequent,

as in the

present token". The transition

z =
3

where

I ), 9

a , i ),r 5 1 5

r

2

6

> 2

1 6

= 1 | W 3 1 6

W

1 1 W 9 1 6

W

7 7

where W

6

= ' the place 1 is empty" or "in place 1 there exists at least one 5 5

SOJE APPLICATIONS OF GENERALIZED NETS IN SCIENCE

109

Fig. 2. 10 token with

the characteristic - an antecedent,

being a subtenu

of the antecedent which is an initial characteristic of the present token",

w = nw . 7

6 The t r a n s i t i o n z <[1 1, !, 11 11 ,, 1l ). ), rr > Z -:
where

where W

3

and W

4

1 77

1

rr = 11 II W W 4 55 11 33 4

W W

are as above.

The transition

8 8 4 4

110

CHAPTER 2

Z 5

= <(1 !, (1 ), r >, 6 9 5

where 1 9 r

: 1 I W 6 1 8

5 where W 6

= "places 1 and 1 are enpty". 3 5 The result

of functioning of this GN is as follows: the tokens

corresponding to the rules of the type exist)

leave the net at place

1 . 8

A I- C and

ASB

K

If all tokens leave

(if they

the net from

place 1 , then in the KB there exist no such rules. 7 The GN checking a given KB for contradictions, i. e. , for simul­ taneous existence of rules of types A I- B and

A I- IB,

is similar to

the GN from Fig. 2. 9, but here the consequent is C = nB. The GN

which checks a given KB

for existence of

two rules of

types A I- B and B I- A also has the form as in Fig. 2. 9 and all notati­ ons are identical without the predicate W = "the place 1 is empty" or "in place 1 there exists a token 1 3 3 which the antecedent

for

(in its initial characteristic) is a con­

sequent in the initial

characteristic of the present token and

the consequent (in its initial characteristic) is an antecedent in the initial characteristic of the present token". As the result of place 1 have initial 6

functioning of this GN the tokens leaving the characteristics - contradicting

rules.

If all

tokens leave the net at place 1 , then the KB is noncontradictlng. 5 Considering the GN, graphic tool

we conclude

for representing

that

the structure

such nets are and functioning

a useful of the

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

ESs. The representation of the ESs by GNs it

will probably become

111

has certain

advantages and

even more popular in future.

This will con­

tribute to an easy transition

from present systems to systems of

the

fifth generation computers. This paper is based on [1]. REFERENCE: [1] E. Dimitrov,

I. KazalarsKi and K. Atanassov.

Generalized net re­

presenting the basic properties of Knowledge base. fic Session of the ligence" Seminar, Sofia, 1989, 12-14.

First Scienti­

"Mathematical Foundation of Artificial Sofia,

Intel­

Oct. 10, 1989, Preprint IM-MFAIS-7-89,

112

CHAPTER 2

§ 2. 7: PARALLEL KNOWLEDGE PEOCESSIHG HOEELLIHG IN EXPERT SYSTEM USIHG GENERALIZED HETS Ivan Hrlstozov

The main task

and

Evgeni Tzolov

in designing expert systems (ES) for application

in fields such as decision support, image trol and others is to

processing,

real time con­

achieve high-speed data processing.

The appli­

cation of the parallel computation principles in the ES as a whole and in the inference mechanism in

particular, is a way to achieve a quick

response. A method for a formalized

description of a parallel

inference

mechanism in ES based on production rules using Generalized Hets (GHs) is proposed

in this paper.

The organization and the interference of

the parallel processes in a multiprocessor system

are being

studied.

The system is built up of specialized processors for parallel computa­ tions (SFPC) of the transputers type [1-3]. The production systems are a popular means for knowledge repre­ sentation.

Each production system consists of three basic components:

a data base, a set of rules (a knowledge

base] and a rule interpreter

(a logical inference mechanism]. The knowledge is represented

by pro­

duction rules of the type IF "condition 1", "condition 2". . . "condition H", THEB "conclusion". The activation of each rule changes the data in the data base. Parallel

data processing in ES on

the basis

of a production

system will be considered as a simultaneous interpretation of a set of rules in a multiprocessor system of SFPC.

The rule evaluation in each

SFPC is sequential. A list of ldentiflcators (names) of the conditions and the con­ clusions

included in the rules

hypotheses to be proved

and a list

are entered in

of identificators of

the process of

the

the knowledge

base formation. The list of hypotheses is a subset of the

list of the

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

conditions' and conclusions' identificators. ledge

base only the rules

113

After forming

the know­

necessary for proving the given hypotheses

are separated. The Knowledge base thus reduced is divided into groups (levels) of

rules, which

can be

activated parallely, i.e.

they

are not in­

formational ly interconnected. Rules whose conditions (the variables on their left sides) are not conclusions of other rules, i. e.

rules that

can be directly evaluated, are referred to as the first processing le­ vel. Rules that depend on at least one rule of the

(i-l)-th level are

referred to as the i-th level (i * 2). Rules of the i-th level may de­ pend on rules of the lower levels.

Only the rules whose

are the chosen hypotheses are evaluated on

conclusions

the last level.

The rules

from a given level are distributed for parallel processing between the processors of the multiprocessor net. For

the case where the number of independent rules

level is less than or

equal to

the number

of a given

of processors in the net,

each rule can be evaluated on a separate processor. When the number of rules on a level is greater than the number of processors, some of the independent rules cannot be evaluated parallelly,

because of the lacK

of resources (processors). If

the net

processors),

of processors is homogeneous

one rule

will be evaluated at

(built up of similar

one and the same time on

the different processors. So balance processor loading in

time may be

chosen as a distribution criterion. Ihe distributed rules (the indivi­ dual knowledge bases) are loaded in the processors. When starting up the inference mechanism, the data base is sent to each processor of the net. When processing the rules of a given le­ vel, each processor changes a given part of it. In order to standardi­ ze the data base

at the end of the rule processing on a

information exchange

between the

SPPC is necessary

values received in each SPPC are necessary to

given level,

(the

conclusion

continue the processing

114

CHAPTER 2

in the other processors of the net). The evaluation process of the rules continues no more

levels. The

until

there are

values of the chosen hypotheses are sent to

the

user (the controlled object), new data (if any) are received from him, and the processing is repeated. When developing sing,

the multiprocessor

the question as to

system for

parallel proces­

the evaluation of its effectiveness arises.

On the stage of designing when the system Is still not physically rea­ lized, this is possible by using the imitation modelling methods. The GNs are Introduced as an effective means for modelling

and

simulating. They permit a detailed investigation, a natural way of re­ presenting

parallel

and asynchronous processes,

receiving temporary

characteristics. Hie operation

of the multiprocessor system

performing

parallel

lnferece in the above stated way may be described by a reduced GN (see App. 1), given on Fig. 2. 11. It has three transitions. Transition Z 1 of the type Z 1

= <(1 , 1

1 , 2

1 , 1 , 1 , 1 , 3 4 7 6

1

), 14

[1 , 8

1 , 4

1 , 9

1 , 5

1 , 6

1

,

the place

1

1

lO

is

!, 13

r >. 1

In the beginning of the GN functioning,

contains

1 tokens,

symbolizing the rules of

the following characteristics:

the Knowledge base.

Each token has

"execution time, "priority". The prio­

rity locates the level of a rule. The rules with an identical priority are processed for one realization of the transition. The tokens from place 1 with the lowest priority pass into place 1 with the charac1 13 teristics - "rule number", "identificator showing whether the operati­ on in the THEN part is performed or not". Tokens with a higher priori­ ty pass in place 1 to without changing their characteristics and 8

are

SCME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

115

Fig. 2. 11 subjected to a subsequent inference all into 1 13

processing.

After terminating

rules are evaluated and the tokens

the logical

from 1 have passed 1

The tokens from 1 go into place 1 , if a history tracing 13 15

of the inference is necessary, or into 1

for a cyclic repetition of

14 the inference. In the beginning of the GN functioning, place

1

(input place)

2 has tokens with initial characteristics "data base identificator" a

"list of

hypotheses we

are interested in".

and

Each SPPC has its own

116

CHAPTER 2

data base ldentiflcator. These toKens pass into place 1 with the cha4 racteristic "updated data base". the next processing level.

Thus 1 4

becomes the input place for

When there are no more levels,

the tokens

from place 1 and 1 pass into Place 1 with the characteristic "final 2 4 9 data base".

In the beginning of

kens whose characteristics

the GN functioning place

1 has to3

are: "number of the corresponding proces-

sor", "numbers of the processors directly connected with it", "identificator of the data basej which is the initial

characteristic of

the

the processing of rules with the same priority ends,

the

token from 1 ". 2 When

tokens from 1 , under the conditions specified below, pass into places 3 1 5

and 1 , with the respective characteristics: 6

sors from which data are received",

"numbers of

proces-

"numbers of processors, to which

data are sent", besides "time for conmunication between the corresponding

processors".

In the case where the number of rules

level is smaller than the number of

processors in

cessors do not take part in the processing with

characteristics

of

a given

the net, some pro-

on that level.

The tokens

"the numbers of such processors" pass

into 1 6

and

have

no characteristics.

operating processors only,

The tokens from 1 , 3

pass into place

"new facts (changes in the data base the corresponding processor

corresponding

1 with 5

to

characteristics

received during the operation of

with the rules of

the specified level)".

So, when passing the transition Z , the tokens from 1 are divided in1 3 to new tokens, the number of which is equal to the number of transmissions/receivings . In 1 7

all processors have the characteristic

"new

SO»E APPLICATIONS OF GENERALIZED NETS IN SCIENCE

117

facts, received during operation". At the end

of the operation the tokens from 1 pass into 1 and 3 lO have no characteristics. The condition r

is of the type 1 1

1 44 1

I

1

false

55

11 66

11 8«

99

1l lO lO

false

false

w 5

false

false

11 1 I 2 1

w

false

1

1

false

W 2

I 3 |

r 1

= 1 I W 4 | 1 11

false W 3

1i

false false

1l 13 13 w 4

w 4

false

false

false

W 4

false

false

false

false

W 4

false

false

false

false

1 I false 7 1

W 2

W 3

false

I I false 8 I

false

false

w 1

false

false

false

false

false

false

w

false

false

ffalse alse

w w 4 4

1 I II II II

w

W 4

1

4

1 || ffalse alse 14 I 14 I

ffalse alse

ffalse alse

w w 55

ffalse alse

where W

= "a new fact is found", 1

W

= "there are processors that will transfer data"; 2

W

: "there are processors that will receive data"; 3

W

- "all rules are evaluated"; 4

W

: "there are groups of rules for processing". 5 The appearance of a token in place 1 activates the transition 9 Z 3

: < [ l , 1 I, II , I , 1 , 1 !, r > , 9 13 11 12 14 15 3

which defines data

exchange with

an external object.

The token from

110

1 9

CHAPTER 2

is divided in two: one part passes into place

ristic

"input data

identificators" and

1 with 11

receives data,

the final user; the other part passes into place 1

characte-

if any, from

with the

charac-

12 teristic "hypotheses" and is sent to the user (controlled object). The condition r

is of the form 3

r

11 11 11 1 I W 9 1 6

3

11

12 12 true

1 II ffalse alse 13 13 II

11

11 14 14 false

ffalse alse

W W 7

15 15 false true true

where W

= "there are new data for receiving"; 6

W

= "a new inference execution is necessary". 7 The

of

transition

processors ready

kens with the

Z is activated if there is at least 2 for

data

same initial

exchange between them

characteristic

one pair

It unites to­

"processor

number".

By

the end of the last exchange operation, all tokens from 1 and 1 have 5 6 passed into place 1 , which is 7

the initial place

for processing the

next level of rules with one and the same priorities. has the form Z = <(1 <(1 ,, 11 ), ), (1 (1 ]], , rr > 2 55 66 77 2 with condition 1 7 1

I W i a r = I 2 I 1 I W 6 I 9 5

Transition

Z 2

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

119

where W

= "the processor in 16 has received all data";

a - "the

W

processor in 15 has transferred the new facts

to the other

9 processors". If a

moment of time t is compared with the moment 1

ting the transition Z , 1

and a moment t

of activa-

to the moment of the

end of

2

exchange with the controlled object (deactivating the transition the difference t - t 2 1

Z ), 3

determines the time for data processing in the

expert system The model developed of

logical

permits on to evaluate the characteristics

inference parallel computers with

a different connection

topology, number of processors, rule distribution,

exchange organiza­

tion (priority of one processor to another) aiming at finding an opti­ mal (according to a specified criterion) version.

REFERENCES: [1] D. Richard, C. Askew and D. Carpenter, Practical parallelism using transputer

arrays, Lecture Notes in Computer Science 258, 1987.

[2] Transputer development system 2.0, IMS D700C, INMDS, 1987 [3] K. Yanev, D. Todorov, N. Avramov and E. Elitzina,

Designing digi­

tal devices with microprocessors, Sofia, Tehnika, 1987 (in Bulg. ).

120

CHAPTER 2

§ 2. 8: GENERALIZED SETS AND DISTRIBUTED DATABASE TECHNOLOGY Alexander Georgiev

§ R. 8. 1. (GN)

Introduction

This article is an introduction to present our

Generalized Net

approach to the Distributed Database Technology

(DDBT). The GNs

should be considered as a possible response to

the problem of tbe de­

velopment of tools for parallel process modelling. The DDBT is a state-of-the-art technology in the database theo­ ry.

The DDBT consists of two basic domains:

Global Intelligence (GI)

with laws and rules governing the Distributed Database (DUB) [1]. The aim of this article is to introduce the foundations

of the

GN approach to the building of the Global Distributed Database Manage­ ment System (GDDBHS)

in the DDBT research area.

the specifications of the ANSI/SPARC wing

project proposed in [2]. Follo­

the current status of the DDBT theory,

to have

Our approach follows

the GDDBHS is considered

two conceptual domains: the DDB as a set of new basic

tural concepts in the DDBT

struc­

as well as a lower conceptual domain,

and

an upper conceptual domain - the GI as a wise manager in a distributed environment. All discussions are on the conceptual level only.

Final­

ly, we propose the architecture of an abstract GDDBMS which integrates all features from the DDBT area on the

conmon basis of the GN

forma­

lism « K

§ 8. 8. 2. Distributed

X

database - basic structural

In the following paper we shall and their consequences of the

discuss three

for organization of

information network.

concepts basic

concepts

the DDB logical structure

They define the data architecture in the

DDB on some layers or schemes, as shown in [1]. The first notion

is fragmentation.

Fragmentation

is a scheme

SCME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

which describes how tables

(relations)

are shared among

121

the network

nodes logically. There are three fragmentation types: horizontal, ver­ tical

and mixed one.

Fragmentation

allows one

to define

different

users' views on the data in the database. The second notion is replication. Replication is a scheme which describes the distribution of different whole tables as copies of data among

the network nodes.

Replication refers to fast local access

to

data as well as to a higher assurance of the network's working. The third notion is allocation. Allocation of

data is a combi­

nation of fragmentation and replication. Allocation describes the fragments with their physical

is a scheme which

addresses on the network.

Allocation is a key element in a project of DDR Allocation has a gui­ ding principle: local availability. The three concepts of the DDB. for

shown above define the layer's architecture

The DDB architecture has

to follow

the basic principles

logical and physical independency of the data.

Therefore,

should exist mappings between the layers. Mapping is facts and

rules for a transformation from one logical aspect

ther logical or physical aspect

of the DDB.

there

a description of to ano­

In that sense, a mapping

could be viewed as a global, fragment at ed, allocated, or local-mapping scheme. it

§ 2. S. 3, Distributed

database - basic

«

logical

concepts

The notion "distributed transparency" is closely related to the DDB architecture. Each layer from the architecture a

conveniently distributed transparent layer.

is associated with

A transparency

DDB means that the user should see that the database user's

node with

its whole entity.

resided

of the at the

The distributed transparency

is

orientated to ensure both logical and physical independency of the da-

122

CHAPTER 2

ta in the DDB. The fragmentation transparency has the highest transpa­ rency level. The Distributed Data Integrity (DDI)

is a feature that ensures

the consistency and security of the data. It is related

to the follo­

wing problems of supporting the activities on the DDB: query optimiza­ tion, concurrency control and transaction handling.

* K

§ 2. 6. 4. Global

It

Intelligence

We consider

GI as

an entity which is most important

for the

the existence of the DDB. A node contains a Database Management System (DBMS). The DBMS represents Local Intelligence (LI) in the DDB system The GI

exists above the LI. GI could

be considered as a

DBMS (DDBMS). The GI manages and controls data, Knowledge about the data

(metadata)

Distributed

the data structure, the row and all

activities of the

overall DDB. The GI is responsible to ensure: (a) Distribution

Transparency

as a structural

aspect as well

as an

activity in the DDB; (b) a DDI

by means of supporting the query optimization,

concurrency

control and the transaction handling. All these activities require the capacity to maintain processing. Parallel processing is related

parallel

to the notion of real-time

synchronization. It is the biggest problem in a DDBMS. For solving the problem it is necessary to have a power modelling tool for behavioural research of a parallel processing

in the real-time situation. The GNs

are the ideal ones in this sense. The GN can behave as

a set of

pro­

cesses at a certain moment of time, each one with its own history, lo­ cal time and constraints. Moreover, the GNs can forecast the behaviour of a stand-alone process, cesses

as a whole,

a subset

and then detect

of related processes and the pro­ deadlock or

wrong situations as

SOME APPLICATIONS OF GENERALIZED NETS IN SCIENCE

well as correct them We consider the

123

GNs a conceptual modelling tool

for a GI above the DDB. We distinguish

some sub-models,

for example:

GN-model for query decomposition, GN-model for query optimization, GNconstraint integrity model, GN-transaction handling model, and GN-conceptual model of the GI (GNGI-model), which is a general one.

« § 2. 0. 5.

«

Conclusions

In this paper we have presented the idea of the GN in the DDBT area. We have reason to thinK that this to

the building of a DDEM5 as a GI in

will

is a new approach

a distributed environment.

discuss the submodels of the GNGI-model precisely

works.

Moreover, the idea of using

DDB area could reflect

application

We

in the future

the GN as a modelling tool in the

the performance of intelligent and power data­

base machines.

REFERENCES: [1] R. Davis, Sharing the Wealth, BYTE, September 1989. [2] The ANSI/SPARC DBMS framework, Information Systemsm, 3, 1978.

Vol. 3, No.

Chapter

3:

SOME A P P L I C A T I O N S OF GENERALIZED NETS IN ECONOMICS, INDUSTRY AND TRANSPORT

Several GH-models describing various processes in areas of eco­ nomics, industry and transport are introduced. Some of them are publi­ shed for the first time (§3.3, § 3.4, § 3.5, §3.7, § 3.8). These mo­ dels

describe the processes from the view of Bulgarian economics, in­

dustry and transport. Some of them give only an idea about the form of the GH-models which can be constructed (e.g. in § 3.3), others descri­ be the processes in a broad outline

(§ 3. 1, § 3.4, § 3.5, §3.7, § 3.

fl). The model from §3.2 and those from §3.6

are more detailed. The

models from §3.7 reflect the first steps made on the global modelling of the NEFTOCH1H Petrochemical Combine in Bourgas (Bulgaria). Therefo­ re,

the GH-models

described here are not

universally

valid

or of

worldwide importance. On the other hand, they do illustrate the possi­ bility for a GH-description of such processes and also for other simi­ lar ones. The models are realized in the frames of the current version of

the program package for GHs and they can be given to those who are

interested in them.

124

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

135

§3.1: MODELLING THE ACTIVITIES OF IHTERHATIONAL TRANSPORT MARKET BY GENERALIZED NETS Antoaneta Kirova

The international

and Krassimir Atanassov

transport market

is a complicated dynamical

system developing under the influence of a number of factors. Its stu­ dy

is connected

aspect

with creating

different models which

give a visual

of the processes and afterwards make it possible

to determine

the ways of improving them in practise. The modelling of economic processes includes retical and

statistical data

The models give an idea about ging

the elements

tasks of

statical which

the interconnection of ways

of a complicated

economic system

the management at a given stage define

modelling:

statical,

modelling

dynamical, is

a number of theo­

concerning their appearance in reality. of arraan-

The gears

a number

expert-imnitational and so on.

characterized

surround the economic process.

by firmly set

and

of ways of

out

The

conditions

An example most typical in this

aspect is the transportation task. The creation of dynamical models is possible with non-stopping processes or phenomena. of

computers

in practise

gives

processes and phenomena which only describe.

This concerns

The widespread use

the possibility of modelling

those

the expert-imnitationa] method can

a number of management processes,

parallel-

going processes, and so on. One of the ways of creating expert-inmitational and simulational models

is by using different types of nets.

The main quality of QNs is that they give not only surement of the tokens, place

but also

the conditions

a time mea­

for going

from one

to another. This creates some prerequisites for simulating re­

al processes and for optimising the existing ones. Here we intend to show two of the processes connected

with the

international market of transport services. The modelling concerns the

126

CHAPTER 3

activities of the forwarding agent and the transporting firm.

» « Let the GU

*

of the forwarding agent be

E

(see Fig. 3.1).

We

c shall assume means

that the tokens

conducted by the

represent the

forwarding agents.

different transportation These tokens possess

the

following characteristics: type of the transportation means, the (cur­ rent) locations, time for covering a certain distance and price of the transportation service. All these are changing in time and in the uni­ verse when passing from one place to another. The forwarding agent's (reduced] GN is as follows: o « E : <(t, 1 , f>, K, , <X, $, b>>, c A where (a) A = fZ , Z 1

Z ), 2

6

(b) the set K of tokens consists of all the trailers and other portation means which the forwarding agent might

use at

trans­ a moment

of time. (c) the time GH's components are as follows: T - a fixed start time-moment, o t - one day, t

- one year. The token's characteristics, which are related

a very

important part

of the generated

to the time are

information in the net.

initial token's characteristics have the form

<"type of the transpor­

tation means", "location"). The characteristic function $ lustrated at

each place after

and for them, the function

$

The

will be il­

its description.

The GH has 12 places

juxtaposes to the

corresponding tokens

SCBE APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

Fig. 3. 1 the following values: 1 - "location of the transportation means", 1 $ - <"new (current) location", "price of passage">, 1 1 - "transportation means sent for loading", 2 $ - <"kind of a good", "price of passage", " time for passage">, 2 1 - "transportation means of sending to the agent's warehouse", 3 $

- <"Kind of a good", "price of passage", " time for passage">, 3

1 - "transportations means of sendinf for re-loading", 4

12T

128

CHAFFER 3

$ - <"direction of transportation">, 4 ] - "transportation means for proceeding to the consumer", 5 $ - <"direction of transportation">, 5 1 - "transportation means for a mixed transportation", 6 $

- <"location", "time for loading operations", "price of passage"), 6

] - "transportation means sent to an intermediate's warehouse", 7 $

- <"location", "time for loading operations", "price of passage"), 7

1 - "transportation means is discharged", 8 $ - <"direction of transportation"), 8 1 - "trailer/road railer on mixed mode of transportation", 9 $

- <"direction of transportation"), 9

1 - "trailer/road railer at the final point", 10 % - <"location", "time for loading operations", "price of passage"), 10 1 - "emergency situation", 11 $

- <"emergency situation"), 11

1 - "goods in the consumer's warehouse", 12 $

- <"location", "time for loading operations", "price of passage"). 12 "Ehe above mentioned places and the characteristic functions re­

lated to them, because he

correspond

to the activity

represents the goods'

of the forwarding

agent,

owners to the transportation firms.

He is concluding the transportation contracts, determines the mode

of

transportation and is used to differentiate other operations connected

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

with

the preparation,

running and

completion of

129

the transportation

process. Components which are anticipated to have an indirect participa­ tion in the model are emitted.

They are t -, t 1 2

and M-transitions'

components, it -, c-, © -, © -, u -, © -, b-GNsJ components. Therefore, L 1 2 K K the described ©? is one that is a reduced from the class A ,A ,A ,11 , c, © , © ,TI , © ,b 3 4 6 L 1 2 K K

E The capacities

of the places

and the arcs

are infinite.

The

transitions have the following forms: Z

= , 1 7 8 12 1 1 7 8 12 where 1 1

r

= 1

1 I true 7 I I 1 I true 8 I 1 I true 12 I

The transition's type of Z

is "v" because it can be fired if a 1

token of

(a transition mean

its input

places

which is already discharged) appears in one

(i.e.

in 1 , 1 , 1 7

ponds to the "sending

8

). This transition corres-

12

of the transition means from

forwarding agent". Z = <[1 1, (1 , 1 , 1 !, r , v(l )>, 2 1 2 3 4 2 1 where

its owner

to the

130

CHAPTER 3

1 2 r = 2 2

1 1 1

I W 11 4 1

1 3

1

W

W 3 3

4

2 2

where W

= "the transportation means is with the producer", 1

W

= "the transportation means is within the agent's warehouse", 2

W

= "the transportation means is sent for loading"; 3

Z 3

= , 2 3 5 6 7 3 2 3

where 1 5 r

= 3

1 2

I 1

W 4

1 3

I 1

W

1 6

1

W

W 6

7

5 W 7

false 8

where W

= "the transportation means is moving from the producer to the con4 sumer",

W

= "the transportation means is for re-loading", 5

W

= "the transportation means is in the intermediate's warehouse", 6

W

= "the transportation means is moving from the agent's warehouse to 7 the consumer",

W

= "the transportation means is for re-loading in the agent's ware0 house". Here

1 and 1 are input and 1 , 1 , 1 2

transition's type is

3

5

6

are output places. The 7

"A", because for its activation it is necessary

to have at least one token in each of the transition's Z means "first stage of transportation". 3

input places.

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

131

Z = <[1 ), tl , 1 ), r , v(l )>, 4 6 S 9 46 where

r = 4

1 6

I |

1 «

1 9

W

W 9

10

where W

= "the transportation means is discharged", 9

W

= "trailer/road railer continuing". 10 Z denotes "end of the first stage of transportation and begin4

ning of the second one. Z 5

= <(1 , 1 ), [I , 1 ), r , A(1 , l )>, 9 4 lO 11 5 9 4

where

r = 5

1 lO

1 11 W

1 9

I I

W

1 4

I 1

W

11

12 W

13

14

where W

= "the goods have arrived", 11 = ~\W

W

12 W

,

11 = "the transportation means have arrived",

13 W 14

= nW . 13 Here

1 and 9

1 are input, and 1 and 1 4 10 11

are output places.

The transition's type is "«" because for its activation it is necessa­ ry to have at least one token in each

of the transitions' input

ces. Z means "end of the second stage of transportation 5

pla­

and beginning

132

CHAPTER 3

of the third one". Z 6

-- < U , 1 ), II 1, r , «|1, 1 )>, 5 10 IE 6 5 10

where 1 12 r

=

1 ] I true 5 I

6

1 | true 10 I Z expresses "completion of the transportation process 6

and de-

livery of goods to the consumer". The proposed agent's

model

shows only

those parts of

the forwarding

activity which require management in time and universe.

is particularly

This

important in the international field, where the deli­

very term needs punctuality in the contacts between the goods and

the

transportation means.

the

The effectiveness is greater in organizing

multimodal transportation abroad, for the processing there is

divided

into different stages (land - river - land, land - see - land, etc. ). > •

«

The transporter's GH E is as follows: n o

■

E = <, K, , <X, i, n A

b>>,

where A = IZ , Z 1

Z ] 2

6

(same as above, but the new transitions are different from the above}; the elements of the set of toKens K also

represent the transportation

means in the transportation firm; the time components of the net are the same as with E ; the token's characteristics in E are analogical c n

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

to E ; c

the initial token's characteristics are from the type:

133

<"type

of the transportation means", "location">. The places of this net denote the following: 1 - "the transportation means is ready for loading", 1 1 - "the transportation means is sent for loading", 2 1 - "the transportation means meets 3 port",

goods from

a multimodal trans-

1 - "the transportation means has got the goods", 4 1 - "the transportation means has given 5 port",

the goods to another trans-

1 - "the transportation means is discharged", 6 1 - "the transportation means is moving loaded", 7 1

- "the transportation means has left the goods to the consumer",

a 1 9

- "an emergency situation",

1 - "end of the second stage of a multimodal transportation", 10 1 - "an emergency situation with a multimodal transportation", 11 1 - "continuing of the second stage of a multimodal transportation", 12 1 - "start of the third stage of a multimodal transportation", 13 1 - "repair/storage of the transportation means". 14 The function $ of the places is similar to the above: $

- <"location", "time", "price of passage")-, 1

$ - <"direction of transportation'^, 2

134-

$

CHAPTER 3

- <"direction of transportation"), 3

$ - <"direction of transportation"), 4 $

- <"direction of transportation">, 5

$

- <"direction of transportation">, 6

$

- <"direction of transportation",

"time", "price of the passage"),

7 $

- <"direction of transportation">,

a $

- <"eniergency s i t u a t i o n " ) , 9

$

- <"location",

"time", "price of passage"),

10 $

- <"emergency situation"), 11

$

- <"direction of transportation"), 12

$

- <"direction of transportation", "time", "price of the passage"). 13 As with E , components which do not participate directly in the c

model will be omitted. This GN is a reduced one from the same class of reduced GNs as the first one. The transition Z

= , 1 6 A 14 1 1 6 S 14

where 1 1 1 I true 6 I r = I 1 1 | true a i 1 I true 14 |I denotes

"the transportation means is ready for taking up goods/out of

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

Fig.

3. 2

usage". Z - <(1 I, 2 1

(1 , 1 , 1 ), r , v ( l )>, 2 3 14 2 1

where 1 2 r

= 2

1 1

I 1

1

1 3

W w 1 2

where w = "the t r a n s p o r t a t i o n means i s working", 1

14 W w 3

135

136

W

CHAPTER 3

= "the transportation means is sent to meet the goods", 2

W

= "the transportation means is sent to be repaired". 3 Z

denotes "the transportation means is sent to meet the goods". 2

Z

= <(1 , 1

3

2

),

(1 . 1 , 1 !, r , v ( l , 1

12

4

5

6

3

2

)>, 12

where

rr

33

= =

11 22 1 12 12

II 11 II II

1 1 1 14 15 16 4 5 6 W W W W W W 44 55 66 ffalse a l s e true true false false

where W

s "the transportation means is moving down a road", 4

W

= "the transportation means is on water/railway", 5

W

= "leaving of goods and moving back for taking the new ones". 6

Z

= <(1 , 1 , 1

4

4 7

),

11 , 1 , 1 i, r , v ( l , 1 , 1

13

7

8

9

4

4 7

11 4 4

1 17 7 II W W 1 6 6 1 I W W 11 66

)>, 13

where

rr

44

= =

11 77

11

I 13 I

1 18 8 W W 7 7

1 19 9 W W 88

W W 77

W W 8

W W

W W

6

7

W W 8

,,

where W

= "the transportation means is moving on", 6

W

= "the transportation means is at a new location", 7

W

= "the transportation means is having an emergency". 8

SOME APPLICATIONS OF (3*5 IN ECONOMICS, INDUSTRY AND TRANSPORT

137

Z denotes "the goods are transported by land to the consumer". 4 That transition aims to model the movement of the transportation means by land in different countries. Z

= < U ), (1 , 1 , 1 J, r , v(l )>, 5 5 10 11 12 5 5 where

r

= 5

1 5

| I

1 10

1 11

1

W

W

W

9

12

10

11

where W

= "trailer/road railer at the end of the second stage of transpor9 tation";

W

= "trailer/road railer having emergency"; 10

W

= "trailer/road railer moving on". 11 Z

denotes

"movement of the transportation means in the second

5 stage of nultimodal transportation". Z = < U , 1 ), (1 I, r , M i , 1 )>. 6 9 10 13 5 9 10 where 1 13 r = 6

1 I true 9 I 1 I true 10 l

The transition Z 6

is of the »-type

because it shows the exis-

tence of at least one token at both its input places. It describes the "end of the second stage of multimodal

transportation and begining of

a third one". Unlike GN E , c

there is a cycle between the transitions

Z and 5

138

Z

CHAPTER 3

here, which corresponds to going in a certain direction. The notion 3

of the transportation means with verse

causes

a change of their locations in uni­

increasing charges for passing to a definite stage and

leads to changes in the characteristics of the tokens. The GN E can n be used for (a) prognosing the expenses of the transportation firm for giving ser­ vice

when knowing the direction and the way of changing

the cha­

racteristics; (b) if the time-parameters related to the tokens are known, the trans­ portation firm can follow their movement and send

them for dispo­

sal of the forwarding agent for a further use and new contracts. The above models

show only a part of the possibilities of GNs.

These models could be used not only for simulation of concrete proces­ ses, but also for management (in combination

with expert systems) and

control in real time. Thi s paper i s pased on [1]. REFERENCE: [1] A. Kirova,

K. Kirov,

K. Atanassov,

Modelling the

transporting market a c t i v i t i e s with generalized nets I z t c h i s l i t e l n a Tehnika i Avtomatizirani sistemi", 66.

(in Bulgarian).

international "Avtomatiua,

1985, No. 6, 61-

SCHE APPLICATIONS OF GHs IH ECCNOHICS, INDUSTRY AHD TRAHSPQRT

139

§ 3. 2: GENERALIZED HETS FOR SIMULATION QF BOOKING SYSTEMS FOR PASSENGER TRANSPORT Sergej Nedev

and

Lllija Atanassova

The automated systems for booking and sale of tickets are ter­ ritorially distributed complexes having a set Toe

of parallel

building of such systems requires great investment.

models

are necessary for a detailed analysis

function of toe complexes,

of

and for correcting

itself. The result of this analysis can be

processes. Mathematical

the parameters

errors in

and

the project

used for choosing an opti­

mal hardware and software for building of the system A

mathematical

instrument for analysing

such systems

is the

simulation based on GNs. A GH represents not only the structure of the process but also the dynamics of the process. In comparison with other net models, GNs are very compact and simple in structure. They provide very detailed information about the simulated processes. In 11] a model of a booking system is described. Here the topic of the simulated processes will be extended. Let us examine the GN mo­ del

of a fragment of a booking system.

The fragment consists of

the

following components: 1. Points for booking - 30; 2. Box-offices at every point, numbered from 1 to 8; 3. The main computer centre of the system (HOC); 1. Telephone channels

for comunication between the HOC and the points

(three logic channels, each containing 5 physical channels). The model gives the possibility of the information about a part of the vehicles to be stored at the points from which they start.

The

channel access is of the type CSHA/CD. The maximum length of a message is 20 bytes. The transmission speeds in the channels are: HCC-point

-

24O0 bit/s;

point-box-office

- 500 000 bit/s.

140

CHAPTER 3

MCC response to a request is 0. 5s. Our GN model is based on the following reduced GN (see § 1. 1> , <X, $>>, where A = !Z , . . . , Z ) is a set of transitions of GN, K is a set of 1 9 Every

system's request is represented by exactly one token in

tokens.

the GN

model. Every token has the initial characteristic of a couple of posi­ tion n

integers (m,n), where m is the number of the

the number

of the box-office

at

the booking

booking point point.

characteristics are exactly the elements of the set X. specifies for every token ct€K the moment

and

These token

The function 6

of time it will enter the GN

through the initial place 1 . It is convenient to considered 1

the cha-

racteristic function as 16 $ = U $, i=2 i v*iere $ is a characteristic function related to the place 1 (the toi i kens

enter

1 with their previous characteristics without receiving 1

such in 1 ). A variable can be specified in the model. Its values cor1 respond to the increment of time for the simulated process. ponent

is not related

global time components

to the global

This com­

time components of the GN.

are not necessary for this

model and

The

that is

y/tiy they are omitted. Let T denote the current value of this variable, T is its initial value, t is the time for the stay of the token 0 a the GN and t i

a in

the time for the stay of a token in the place 1 (1 <. i i

i 16). The places

and

the characteristic

have the following meanings:

functions

related to them

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

1

141

- stands for box-offices. In the box-offices the users requests are are created, shown as tokens of the GN.

1 - stands for points of booking. All requests are collected 2 point. $ - specifies the value 2

X + t a 1

as a characteristic of

the

the tokens,

where X is the final a

characteristic of

0. 002 s .K,

is a random number for which 1 S K 5 9 (the

where K

number 0. 002s

the token

at

a

and

t = 1

is the duration of a single transmission between a

box-office and a booking point). 1 - stands for the queue of requests, waiting for sending through 3 the telephone channel between the booking point and the MCC. $ 3

- specifies the duration of the tokens' stay in place 1 . 3

For 4 i i i 8: 1 - stands for sending i channels

of

the request through

$ - specifies the value X +t + t i a 2 3 kens, where t

as

= 0. 1 . K, where K

one of the telephone

a characteristic of the

to-

is a random number, 0 i K <. 4.

2 It is assumed that in 70Z of the cases are in 0 i K i 1. The num­ ber 0. 1

is the duration in seconds of

the sending of

a request

through the telephone channel, t

1 9 5 9

= 0. 1 . (Q + 1), where Q is the 3 number of the simultaneously transmitted requests (4 <. i < 8). - stands for request reception in the MCC of the system - specifies the duration of the tokens' stay in the place 1 . 9

1 - stands for request processing in the MCC. 10 $ 10

- specifies the value X + t as a characteristic of the tokens, a 4 where t = 0. 5 s. (It is assumed that one request is processed 4

142

CHAPTER 3

In the average time of 0. 5 seconds). 1 - stands for the queue of requests, 11

processed in

the booking

point itself. $ 11

- specifies the duration of the token's stay in the place 1 11

1 - stands for request processing in the booking point. 12

$ 12

-- $ . 10

1 - stands for cashier reception of the system response. 13 $

- specifies the value

X +t +t as a characteristic of the tokens, a 1 5

13

■where t = L, L being a random number equally distributed in the 5 the interval 60 - 120s. 1 - stands for the end of request processing. 14 1 - is analogous to 1 15 13

$ 15

=$ . 13

1 - is analogous to 1 16 14 The capacities of the different for 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 2 3 9 11 13 14 15

places are:

for 1 - infinite, 1 - 240, for 1 - 30, for 1 16 12 10

1, for 1 , 1 , 1 , 1 , 1 - 24. The capacity of each arc of 4 5 6 7 8 sitions is 1,

the tran-

e. g. every element of the capacity matrix is equal to 1

and that is why this component will be omitted in the transition defi­ nition below.

Besides all the transitions are disjunctive and that is

why this component will be omitted too. The formal description of the GN transitions of aring in mind that this ON is reduced) is Z 1

= , 1 2 1

Fig. 3. 3

(be­

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

143

Fig. 3. 3 where 1 is the input place for Z and 1 is the output place for 1 1 2

Z, 1

the transition condition is 1 2 r 1

= 1 I II

W 1

where W

= "in the

GN

does not exist any other token with the same initial

1 characteristic".

144-

CHAPTER 3

Z = <(1 Z <(1 ,, 1 ), ), (1 (1 ,, 1 ), ), rr >, >, 2 22 10 10 33 11 11 2 where 1 33 rr

-- 11 22 22

II W W 11 22

1 11 11 W W 3 3

1 II W W 10 II 4 10

false false

where = "d = 0" & "T i X

W 2

= "3 = 1" & "T i X

W

+ t ",

a

3

a

1 + t ", 1

where d is an indication for branching at the transition Z . 2 W = "T i X + t ".

3

a

4 Z 3

= , 3 4 5 6 7 8 3

where the r ' s p r e d i c a t e s c o n t a i n the e s s e n t i a l of t h e channel 3 type CSMA./ CD. Z = <[1, 4 4

I, 5

1 , 1 , 1 I, 6 7 8

|1, 9

where

r 4

1 9

1

1 I 4 |

W 4

W 5

1 I 5 | = | 1 I 6|

W 4

W 5

W 4

W 5

1 I 7 |

W 4

W 5

1 I 8 1 8 1

W 4 4

W 5 5

where

w 4

= "a - o" & "T i x a

+t

+ t ", 2

3

15

1 15

), r >, 4

access

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

= »S - ln

W

& "T i X

5

+ t a

2

+ t ", 3

where 0 is an indication for branching at the transition Z . 4 Z = <(1 ), (1 ], r >, 5 9 10 5 where 1 10 r

= 1 I true ; 5 9 5 9I I Z = 6 11 12 6 6 11 12 6 where

r = 1 I 6 11 I

1 12 12 W 6

where W

= "in 1 6

there are no tokens with the same characteristic m"; 12 Z

= <(1 7 7

), (1 12 12

13 13

), r >, 7 7

where

r = 1 I 7 12 I

1 13 13 W 7

where = "T i x

w 7

a

+ t ": 4 Z

= <(1 8 8

13 13

i, (1 ], r >, 14 8 14 8

where 1 14 r = 1 I 8 13 8 13 I I where

W 8 8

14-5

146

W

CHAPTER 3

= "T i t +t + X ". z fl 1 15 5 a Z = <(1 9 15 16 9

where 1 16 r = 1 I A 15 a is Ii

W . A a

The described model shows the basic points of the structure and the function of an automated system for booking and sale of tickets in the passenger transport. This model can be used for two principal aims: 1. When the main parameters of the system

are specified

productivity of the MCC, loading of the system, of the channels,

by this model the following

such as

transmission capacity characteristics

of the

system can be defined: a) average time for processing of a request; b) maximum time for processing of a request; c) the part of processed requests for a fixed period of time; d) the time for processing the requests in

every

element of the sys­

tem; 2. When the loading of the system is specified,

the productivity

of the elements can be defined so that the average time for processing of a request does not overrun a certain value. 3. The number of the processed requests for

a defined

period of

time and others. The suggested model can be developed, if necessary, for getting extra information about the simulated processes. This paper is based on [2]. REFERENCES; [1] S. Nedev,

K. Atanassov,

An automated

booking system modeling by

generalized networks, Automation, Computer Engineering and Automa-

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

14-7

tion Systems, 1989, No, 6, 17-20 (in Bulgarian). [2] S. Nedev,

L. Atanassova,

zervation systems for

Generalized nets for simulation of re-

passenger transport, First

the Math. Found. AI Seminar,

Sci. Session of

Sofia, Oct. 10, 1989,

MFAIS-7-89, Sofia, 1989, 45-49.

Preprint

IM-

148

CHAPTER 3

§ 3. 3: GENERALIZED NETS IN MODELLING OF PUBLIC TRANSPORT Katja Stefanova

In this section,

public transport is modelled by means of the

Generalized Nets (GNs). "A transport system"

means

the set consisting

of a transport

net in a given urbanized region, all vehicles and passengers, her

active or passive

and ot­

participants in the public traffic of the same

region, A transport net is characterized by

its geometry, topology and

organization. Its aim is to realise vehicle

or pedestrian

connection

between al1 points which are to be connected in a given region. We shall (buses,

describe transport nets

trams,

public

transport

trolley-buses etc.) only. Such a transport net can be

characterized by

a directed graph

stations. Each pair of vertices a given route

involving

G

whose vertices

represent

the

representing neighbouring stations of

is connected with

arcs directed according to the traf­

fic. In the sequel we shall use the GN

for transport systems model­

ling. Let V = |v, v 1 2 be the set of vertices, and

v ] P

A = |a, a 1 2

a ) q

be the set of arcs of the graph G. To each vertex

v € V

and its cor­

responding input and output arcs a', a', . . , a'; a", a", . . . , a" 1 2 i 1 2 j 1 i H

p, l i j i q

where

(see Fig. 3.4), we shall assign the transitions

2'and Z" from the GN (see Fig. 3.5). In view pairs

of the results of

§ 5. 1 in App. l,

of transitions corresponding to

the vertices

the union of

all

of the transport

SCME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

Fig. 3. 4-

Fig. 3. 5

149

150

CHAPTER 3

net represents a GN. This GN contains transitions of the two types gi­ ven

above and two kinds of tokens corresponding

the passenger stream

to the vehicles

and

The tokens of the first kind will run over pla­

ces of the type 1', 1" and 1 and the second kind will run over places 2 of types 1"', 1"", 1 , 1 , 1 . i 2 3 The GN will also contain global time parameters such as an inio T, an elementary time-step t <e. g. a minute, for "ro­ the time step may be larger), duration of the model func-

tial moment ugh" models tioning t

(hours up to twenty-four hours).

of the GN has hicles,

the following

The first kind of tokens

initial characteristics: type of the ve­

registration numbers, numbers of the line.

kind of token,

the number of

the passengers

As for the second

boarding

a vehicle

is

its initial characteristic. The tokens of the first kind are in the GN at the moment T,

whereas the tokens of the second

kind are in the GN

during its functioning through the places of type 1 . In

the traffic,

4 the tokens of the first kind can be characterized necessary

also

by

"the time

for a transition from an ex-station to the current station"

(in places of type

1J)

as well as by

"the sojourn time in

the last

station" (in a place of type 1"). The tokens of the second kind can be characterized by

"the number of passengers

vehicle" (in a place of type 1"") and

who are currently

in the

"the number of passengers

get­

ting off the vehicle" (in a place of type 1 ). The tokens of the se1 cond kind do not possess any characteristics in a place of type 1 . 3

A transition of type Z' has the following form: Z'

=
1' 2

v(A(i', i"), 1 1

1', i A(r, 2

1", 1

1" 2

i») 2

1"], i A (

r, i

11 , 1 , 1 2 !">)>, i

1 i, 3

»,

x,

r\

x,

SOME APPLICATIONS OF <2>!s IN ECONOMICS, INDUSTRY AND TRANSPORT

151

where a) 1', I',..., l' 1 2

and

i

1", 1",..., 1" 1 2

correspond to the arcs between

i

the vertices of the graph which are connected with the vertex v; b) 1 is related to passengers getting off the vehicle (the tokens re1 ceive in it the numrber of these passengers); c) 1 is related to the presence of the vehicle; 2 d) 1 is related to the presence of passengers in the vehicle (the to3 Kens receive in it the number of these passengers); 1 1

r'

=

1 2

1

1' | false 1l

true

false

1' I false iI I 1" | W 1 1 1

true

false

1" I j I

W

false

3

W 2

false

W

1

2

where - "there is at least one passenger getting off the vehicle";

W 1 W

= "there is at least one passenger continuing the trip". 2 A transitions of type Z" has the following form;

Z" = <[1 , 1 , 1 ), fl"\ 1"' 2 3 4 1 2 x, A(1 , v(l , 1 )>>, 2 3 4 where

1"', 1"". 1"" i 1 2

1""!, *. *, r", i

152

CHAPTER 3

a) place 1 is related to passengers boarding the vehicle 4

(the tokens

receive in it the number of these passengers); e) the transition condition is

I I r" =

1"' 1

...

1"' j

1"" 1

...

1"" J

w

...

W

false

...

false

2 1 3 I 1 I false 3 1

3

1 | false 4 |

...

false

W 3

...

W 3

...

false

W 3

...

W 3

■where W

= "the next route station of the vehicle". 3 All transitions have equal priorities. The GN described above is a reduced GN of the type

A,A,A,Tl,e,9,b 3 4 6 A 1 2 T. We conclude that within the framework of the model we can

spe­

cify the following main characteristics: priority of the stations (for the places of type

1 ),

capacity of the places of type 1

2

(number of

2

vehicles stopping at a given station) and the priority of the vehicles (tokens of the first kind). The fornullation of an analytical model of the considered tran­ sport system, particularly the analytical

part of the given GN model,

is related to the influence of factors of various nature. Some of them are:

settings and movements of the population,

presence of different

social and age groups with the appropriate requirements

for security,

comfort, costs, and travel time. These factors affect the determinati­ on of the initial

parameters of the second kind of tokens

passengers waiting for a vehicle at a given station,

(number of

with preferences

regarding the type of vehicles, current number of passengers, etc. ).

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

153

Different characteristics of the transport system can be obtai­ ned

by simulating the model built by means of a (34.

It makes it pos­

sible to use the GN model as a tool for transport process

management.

Furthermore, it enables optimization of the vehicle routes the

traffic process and passenger streams.

simulating

It is easy to obtain

the

average and maximum time duration of a given route. In the case of any changes, the time schedule and the correspondingly adjusted streams

can be fixed.

On the basis of the obtained results,

passenger strati­

fication of the transport net and appropriate estimation of the travel costs can be determined

154

CHAPTER 3

§ 3. 4: GENERALIZED HEX MODEL OF ACTIVITIES OF A BUS-DEPOT Favlin Gyurov

An example for using tion

toe Generalized Hets (GHs) for a descrip­

of some transport situations is introduced.

It illustrates

the

solving of the following problem Suppose a public coach agency uses seats are Known,

n

coaches whose numbers of

when the coaches are not in use, they are in a gara­

ge. There is a service for maintenance work. The coaches leave for the routes they

from a bus station.

take their own daily

Every morning, before leaving the garage, time table.

At the end of

their workday,

they go back to the garage. This process

will be described below by means of a GH and

the

loading of the lines and coaches will be reported. The GH has a graphical structure as shown in Fig. 3.6. Every coach is represented by

a token in the GH.

when a coach

is on a route,

the token corresponding to it goes into the GH. At the

time -moment T,

all coaches are

bus-tokens BT i

in the garage,

so at that moment all

(1 <, i i n) are in place 1 . If any coach is out of or3

der, it has to be

repaired so

the corresponding token goes

to place

1 through place 1 and the transitions Z and Z . when the coach is 13 15 2 3 in

order again the corresponding token will

come back

to place

1 3

through place in

1 and 1

the transitions Z

and Z . If any coach located 4 1

the garage is in order and it is time

to go on a route,

the cor­

responding token leaves place 1 and goes to place 1 through place 1 3 7 5 and the transitions Z 7

and Z . 5

If there is much time before

the next

travel of the coach, it goes to the garage and the corresponding token

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

155

Fig. 3. 6

goes to place 1 3

through place 1 and the transitions Z and Z 2 6 1

The

situation is the same when its travel is completed. However,

if a coach

is damaged and has to go on the route, or

it gets damaged while travelling,

it cannot do the next planned

tra­

vel. So it has to go for servicing. Then the corresponding token

goes

to place 1 that

But the travelling will not be done. So it is necessary

another coach

takes up the next travel.

This is why there is a

156

CHAPTER 3

private token in the GN. At the time-moment T it is in place 1 . This 12 token (which we will mark below by OT) gets as characteristic the next travel which the damaged

coach would to do. Then it goes to place 1 . 4

In this case, there is a choice for an optimal coach from those in the garage. The corresponding token goes to place 1 and the goes to place 5 1 16

The bus-token goes to place 1 7

through place 1 8

and

and the transitions Z

that can accept the order,

5

OT comes back to place

and Z . 8

OT waits in place

If there is no coach

1 . It is possible that 4

at that time-moment another coach come out of order. the transition Z 1 9

9

1 12

and its copy comes back in place

So OT 1

12

splits in

through place

and the transition Z . It is useful if the transitions 8

Z

8

and

Z 9

work more often than the others. Bus-tokens have only one (current) characteristic with the form: "<seats, maintenance, travels>", where seats is a number of sites, maintenance = , where repairs is the set of repairs of the coach up to the moment, travels = ( / t € IT, T+t 1) OT can have a characteristic of the form (see above) "<seats, waiting-time, travels">. Initially, it is in place 1 . When it comes into place 1 , 12 4

it gets a

new characteristic whose seats component is the number of sites of the damaged coach, its waiting-time component travels

component is

is the previous one and the

the set of travels which

when OT comes into place 1 , 9

this coach

would do.

it doesn't change its characteristic. In

SCME APPLICATIONS OF GNs IN ECONOMICS , INDUSTRY AND TRANSPORT

place 1 it gets new characteristics - the previous seats 16 components

157

and travel

and the waiting-time characteristic is the duration of

is

stay in place 1 . 4 Every bus-token gets as

a first characteristic

its

number of

seats and its daily time table. The maintenance component of its first characteristic is

<0, 0>.

When it goes to place

characteristic. Travel component reduces to

1 it changes 13

its

the set of travel fulfil­

led. The current characteristic of the token includes information aboit gets

ut it. When a token comes in place 1

a characteristic - the

1 previous last

of

The nunfcer of seats

travel component.

can be changed

the repair component, and the duration

the repairs join

The of

these repairs adds to the current durat ion of maintenance. When

a token comes into place

1 it changes 14

the passengers

(TIME) characteristic. A bus-token that comes into place 1 can append 3 new travels to the travel component of its characteristic. Z

1 1 = >■ 1 2 1 1 2 3 1 2 1 1

where 1 3 r

= 1

1 l 11 l 1 l 21

true true

1 3 M

= 1

1 I 11 1 1 1 21

n

' n

15&

CHAPTER 3

2 2 Z = <(1 , 1 1, 11 , 1 , 1 ), t , t , r , M , v(l , A(1 , 1 ))>, 2 3 4 15 5 16 1 2 2 2 3 3 4 where

r = 2

1 3 1 4

1 15

1 5

1 16

I W I 1 I I false I

W 2

false

false

W 3

where W

= "the coach is out of order"; 1

W 2

= iW & ("the moment for going to the station has come" or 1 is an OT

in 1 " 4

and

("the coach has no other travel"

("there or

"the

travel of OT can be done before the travel of the bus-token") and "this coach is optimal about the number of seats"); W

= "the order is taken up by the bus-token"; 3 1 15 1 M

= 2

Z

= <(1 3

I n 3 | I 1 1 0 4 I

1 5 n 1

3 3 ,1 ,1 !, (1 ), t , t , r , M , v(l , 1 , 1 )>, 15 10 11 13 1 2 3 3 15 10 11

where 1 13 r r

3 3

= =

1 I 15 15 I I I I 1 I 10 |

true

1 I 111 1 I

true

true

SCME APPLICATIONS OF GNs IN ECONOMICS,

INDUSTRY AND TRANSPORT

1 13 1 M = 3

I 15 I I 1 I 10 I

n

1

n

n

I

11 I Z = <[1 ), |, 4 13

4 4 [1 ]!,, t , t , r , , M ,, v ( l >>, 1 1 2 4 4 13

where 1 r

=

1

4

| 13 I

1 true 1 13 M = 4. 4 Z = <[1 , 1 , 1 ), 5 5 6 16

1

I i13 n Ii

n

5 5 (1 , 1 ), t , t , r , M , v ( l , 1 , 1 )>, 7 8 1 2 5 5 5 6 16

where

1I r

= 5

| 5 I I 1 | 6 I 1 | 16 I

1 M = M 5

5 1 66

1

I 1 I I II

1 16 16 II

0

1 7

1

true

false

true

false

false

true

1 7

I

n

0

n

0

8

a

1

159

160

CHAPTER 3

6 6 Z : = c|) (|) I, II !1 , 1 , 1 ), t , t , r , M , v(l )>, 6 7 10 2 11 1 2 6 6 7 where

r = 6 6

1 7 7

I 1 1

1 10 10

1 14 14

1 2 2

W

W

W

1 1

4 4

5 5

where = iV

W 4 W 5

& "it is time corresponding to the token coach to travel", 1

= iW & 1

("there is nuch time before the next travel" ach's work day is finished" ),

M

: 6

-
Z 7

1 | 7 I

1 10

1 14

1 2

n

n

n ;

7 7 ), fl , 1 ), t , t , r , M , v(l )>, 14 11 6 1 2 7 7 14

where

r

= 7 7

1 14 14

l I I I

1 11 11

1 6 6

W 1 1

w W 6 6

1 11

1 6

n

n

where W 6 6

- nW & "it is the end of travel", 1 1

M

= 7

1

I 14 I

;

Z = <[1 <[1 ,, 11 ]], , fl fl ), ), tt ,, tt ,, rr ,, M M ,, vv(l ( l ,, 11 )>, )>, 12 8 88 88 9 9 12 11 22 88 88 8 9 9 where

or

"the co-

SCME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

161

1 12 1 r

=

a

a 1 9

I I i I I

true true

1 12 M, M, = = a a

1 I 8 I a i i i 1 I l i 9 I 9 I

1

l1

9 9 Z --- , )>, 9 12 12 2 9 1 2 9 9 12 where 1 2 r

=

1

9 9

l 12 I 12 I

1 9

W

true 7 7

where W

= "there is a coach which is damaged"

&

"this coach

7 vels",

M

= 9

1

l 12 I

1 2

1

1

1

9 .

For t h e GN-components t o be v a l i d , i A

|Z | = i f o r i

n(l) L 4 12

S 9,

= n ( l ) = l; for all other places: TT (1) = 2, 9 L

) - 2;

c(l

I $i

for all other places: c(l) - n,

has to tra-

162

CHAPTER 3

i t

o + 2. t , if i £ (1, 2, 3, 5, 6, 7)

1 i i 6 (t ) ) = s 1 1

i o t + C. t , if i = 4 (C = const 2* 2), 2), 1 i t

o + t , if i € [», Eft, 9)

1 It means that the private tokens go faster than others.

In or­

der to maintain the works character it is possible that the transition Z works rarely. 4 i i i o © (t ) = t = t (1 <, i i 9). 9>. 2 i1 2 2 i The numbers t (1 i i £ 9) must be large enough so that all to2 kens can go through transitions K = (BT / 1 i i <, n| tl iOT), i IT (a) K

- 1, for every a € K,

0 (a) = T, for every a 6 K. K All tokens enter the GN at the starting time-moment of its work. T

is the time-moment of the start of workday,

o t = 0. 5 minutes

M

example); T + t

is the time-moment of end of a workday.

(for

SCHE APPLICATIQHS CF GNs IH ECCHCHICS, INDUSTRY AND TRANSPORT

163

§ 3. 5: GEHERALIZED HET MODEL CF THE ACTIVITY CF AN AIR-TRAFFIC CCKTROL CENTER Stanislav Fistroanov

As an example of the description of real processes with Genera­ lized Nets

(GHs) we snail examine a GN-model of

air-traffic control center consists of area fixes

the following:

for landing,

the

activity of an

In general,

this activity

when an airplane is coming to the airport

it comes under

the control of

a dispatcher,

who

an air-corridor where the airplane can fly a round until a run­

way is free. airplanes area,

at an airport.

At the sane time,

which have

there must also

just flown

off until

we shall examine an airport with

n

they tracks

can be used for both landing and for flying off), fic is being North (1),

controlled

be air-corridors for leave

the airport

(each one of which where the air-traf­

by 4 dispatchers - one for each direction:

west (2), South (3) and East (4). All tracks are used by

all dispatchers at the same time. Every dispatcher can control up to m airplanes at any one moment, while in special situations for short in­ tervals of time this number can be increased to m + k, where fixed natural number, i.e. airplanes

waiting for

k

is a

the normal capacity of the airport is

landing at any one moment,

when

4.m

the number of

waiting airplanes is less than 4. m, this means that there are free aircorridors. from

We shall

the same way

corridor

examine the airplanes

which have just flown

as those waiting for landing - each takes

from the moment of flying off till the moment of

off

one air-

leaving the

airport area. Let us examine a GH with 7 transitions and 23 places for move­ ment of airplanes,

and a separate transition Z with 0

tokens moving, carrying managing information

2 places,

whit

(see Fig. 3. 7).

The separate transition Z has two input places 1 and 1 which 0 0 1

164

CHAPTER 3

are also

the output places

for it. In each of them, there is a token

which passes the transition; when it is activated, it renews its tran­ sition and goes back to its starting place, i. e.

the index

matrix of

the transition Z is 0 1 0

1 1

1 l true 0 |

false

1 | false 1l

true

There is only one token

a with the current characteristic (and

a without the other ones) x

= "", 1 2 n

where

a

is the

number of current free air-corridors of the whole airport at the mo­ ment, a , a , . . , , a € 1 0 , 1), a = 0 means that the i-th track is oc1 2 n i cupied at the moment, and a = 1 means the opposite situation (1 <, i i i n). In the place 1 there is only one token - p, which has the cha1 P racteristic x - "", where b is a natural number for 1 2 3 4i which

-k <, b i

i m, and b i

is the number of

free air-corridors under

the control of the i-th dispatcher (1 s i $ 4|. The set S with elements s is the set of tokens which represent j the airplanes in the airport area. set (1,2

Their number is an element

4. m) and they are transferred to the other section of the

GN. Each of these tokens has as an initial characteristic its fying data 1 4 re:

of the

(type,

identi­

nationality, number of course, etc. ). In the place

j j j receives as new characteristic "" 1 2 3

(1 i i £ 4. m), whe-

SCME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT j - c e (1, 2, 3, 4)

determines the direction from which the airplane

comes; for airplanes which determines

have just flown off, this characteristic

the direction to which the airplane

inside the ®*,

165

for airplanes

waiting to land,

will be travelling; this characteristic

carries the appropriate information; J - c is a parameter corresponding to the time during which the airpla2 ne can fly with its fuel;

for the airplanes just taking off

it al­

ways has the maximum value;

J - c € {0, I) and it is equal to 0 for an airplane waiting to land and 3 to 1 for an airplane just taking off. Token s (1 <, j £ 4. m) can enter the GN through three input j places 1 , 1 and 1 . Every token entering through 1 1 2 16 2

corresponds to

the airplane, coming to the airport

If the airport

capacity is not exceeding

(i.e.

area for landing.

a pr x i- 0), 1

then the token moves to

place 1 (where it receives the above characteristic) and (simultaneo4 a usly the token a receives as a new characteristic such one with pr x 1 decreased by i

(the airplane is accepted in the airport).

Otherwise,

the token moves to place 1 , which is an output place for tokens

cor-

3 responding to airplanes (the airplane is directed to another airport). The transition condition of Z

is 1

r

= 1

where

1 3 1 I W 2 1 1

1 4 W 2

166

W

CHAPTER 3

a = "pr x * 0", 2 1

y/

--

1

-iw .

2 A token

s j

which has entered the GN through the second

input

J

place (1 ), corresponds to an airplane just taking off 15

and has

c = 3

1. To move through the transition Z , it is necessary to satisfy the 3 a a condition "pr x * <0, 0, . . . , 0>" & "pr x * 0" (i.e. at 2, 3, . . . , n+1 1 least

one track mist b e free a n d also at least o n e air-corridor for

leaving the airport mist b e free).

B e s i d e s a token leaving place

1 15

can m o v e only to place row

1 16

f o r assignment to a dispatcher,

of the index m a t r i x of the transition Z

corresponding

i. e. the t o place

3 1

contains the value

"false" in every position, so does t h e c o l u m

15 corresponding t o place

1 , w i t h only 16

o n e exception:

the predicate

a related to the places 1 and 1 is "pr x i- <0, 0 0>" 15 16 2, 3, . . . ,n+l a & "pr x i- 0". When a token moves from place 1 to place 1 , the to1 15 16

a ken a receives a new characteristic: pr x decreases by 1. 1 Itie transition Z

stands for the distribution of

airplanes to

2 the dispatcher.

A token s moving through this transition, regardless J

of the place

where it is coming from, goes to place 1 (the corres4+i j ponding airplane is distributed to the i-th dispatcher), where i = c . 1 The transition condition of Z

is 2

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

11 55 11 33

rr

= = 22

11

11 66

II W W 11 33

11 77

167

8

W W 44

W W 55

W W 6

11 II W W 17 II 17 33

W W 44

W W 55

W W 66

11 II W W 18 II 33 18 II 1 1 II W W 19 I 19 I 33

W W 44

W W 55

W W

W W 44

W W 55

W W 66

1 1 II W W 220 0 II 33

W W 44

W W 55

W W 66

11 II W W 16 II 16 33

W W 44

W W 55

W W 66

66

where W

j = "c = i" for 1 <. i i 4. 2+i 1 Transition Z 3

1 . 14

contains two output (of the GN) places - 1 and 13

Tokens, moving to place 1 14

correspond

to airplanes Just flown

off, which are leaving the airport area under the dispatcher's guidan­ ce. A toKen s can move to place 1 , if it cones from place 1 for j 14 4+i j a 1 i i £ 4, and if c = 1 . Then pr x 2 1 Tokens

and

p pr x i

moving to the other output place 1 13

each increases by 1.

correspond to airplanes,

which are landing at the moment. A token s can move to place 1 if j 13 j « if it has c = 0 and "pr x ^ <0, 0, . . . , 0>" (at least one 3 2, 3, ...,n+l a p track is free). After this, pr x and pr x each increase by 1, where 1 i i

d e p e n d s o n t h e G N - p l a c e w h e r e t h e t o k e n comes /' j

- 4,

f or 5 i j

\ j

- 20,

f o r 21 S j

<. 6 i

24

from

168

CHAPTER 3

1 and 7

1 8

1 and 23

respectively, with the exception that in places

1 24

1 , 1 , 21 22

tokens corresponding to airplanes just taking off cannot

j appear. If the equation c = 0 (airplane waiting for landing) is valid 3 for a token s , it j (1 i x i 4), and

comes from place 1 (i £ I i 4) or place 1 4+i 20+i cc pr x - <0, 0, . . . , 0> at this moment, 2, 3, . . . ,n+l

moves to place 1 (the corresponding airplane 8+i corridor for waiting for a track to be free, der

the control of the same dispatcher).

that the transition condition r

receives

a free air-

the airplane staying un­

From the above,

it follows

has the form 3

1 99

= 3

10 10

1 5

I 1

W 7 7

false

1

I false false 1

W

6

r

1

1 11 11 false

12 12

1

1 13 13

false

W

false false

W 7

1

false

false

| false I I 1 I W 21 I 7 8

I false

22 I

false false

W

false W 7

W

W 8

W 9 W

false

8

9

false

W 8

W 9

false

W

false

false

W

W

false

false

W 7

false

1 I false 24 I

false

false

W 7

1

false

false

8

9

W 8

W 9

W

W 8

false

false

false false

9 false

W 10

where a n+1

false

false

1 I false 23 I

x

false 9

false

7

I false 15 I

false

8

W

16 16

9

W

7 false

W = "pr 7 2, 3

1 14 14

8

1 I false 7 |

1

1

j = <0, 0, . . . , 0>" & "c

= 0", 2

it

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

W

a x * <0, 0

= "pr 8

169

j 0>" & "c

= O",

2, 3, . . . , n+1

2

j

W

= "c 9

=1", 2

W

a x * <0, 0

= "pr 10

a 0>" & "pr x * O".

2, 3, . . . , n+1 The transitions

1 (1 & j £ 4)

Z

are completely analogous

to

3+j one another. The

j-th of them corresponds to the functioning of the j -

th dispatcher. Because the airplanes come in from the j-th or fly

Then under his control and at some b

direction,

out in the j-th direction, the j-th dispatcher certainly gets.

= pr x

j

< 0).

moment he can be overworked

(then

That is why when a token moves from place 1

J

(the

a+j

corresponding airplane is waiting for landing under the control of the j-th

dispatcher)

through the transition Z

transition condition,

if b

£0

. After a check 3+j

of the

the token moves to place 1

j

(the

20+j

corresponding airplane continues waiting under the control of the same dispatcher),

else the token moves to place 1 and receives as new 16+j j characteristic the last one but with c = s, where b = max b (the 2 s liti* t corresponding airline will be transferred to the control of a new dis­ patcher

who at the moment

control

the smallest number of airlines).

(1 i j i 4) is

The transition condition of Z 3+j

1 16+j r

= 3+j

where P W

= "b

11

j

: pr x

j

< 0",

1 fl+j

I W I 11

1 20+j W 12

170

CHAPTER 3

Fig.

3. 7

SCME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

W 12

171

= nW . 11 All the transitions and places in

priorities;

the tokens in places 1 , 1 2

the described 1

3

are

GN have equal ordered

as to

24

their second current characteristics in ascending order. The GN starts functioning at fixed moment

a fixed time-moment

T

and it functions until another

T+t . It works tact by tact with an elementary time-step

o t (=15 sec. , e. g. ). All transitions are activated at every tact. All arcs have infinite capacities. All transition types are disjunctive.

m

CHAPTER 3 § 3. 6: GENERALIZED HET MODELS PQR FLEXIBLE HAHUFACTURIBG SYSTEMS Maria Stefanova-Pavlova

§ 3. 6. 1

and Krassimir Atanassov

Introduction

Flexible Manufacturing Systems (FMSs) [1-58] are an attempt to reconcile the efficiency of the production line with

the flexibility

of the job shop in order to satisfy a versatile demand at low cost. Generally two Kinds of flexibility are distinguished. The longterm flexibility corresponds to the possibility product group in the manufacturing

of introducing a new

systems during

its operation and

with little effort. The short-term flexibility corresponds to the pos­ sibility of handling a large variety of product groups at a given ti­ me in the manufacturing

system [7]. In order

to meet these require­

ments, an FMS is made up of (a) a set of flexible machines, (b) an automatic transport system, (c) a sophisticated decision-making system to decide at each instant what has to be done next and on which machine. Flexible machines have

the capability of performing

operations. They have an automatic tool storage with

various

a retrieval sys­

tem and the machine programs can be downloaded. A n automatic transport system is required to transport the parts for the next operation. The decision making system organizes the production schedules and synchro­ nizes machine

utilization. This system is very important

for solving

problems such as idle time minimization and manufacturing without pe­ ople, which appear in organizing the work of FMSs. A FMS is structured cell

into

is an elementary system

some local storage facilities

flexible manufacturing

cells.

Each

consisting of a flexible machine tool, for tools and handling devices

such as

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

robots in order to transfer parts and

tools

173

between the cell and the

global transport system [7]. The FMSs are a class of automated manufacturing systems. An EMS comprises processing modules or machines linked by a material handling system, all under central computer control (cf e.g. [3]). The flexibi­ lity of such a

system is therefore dependent on its components' capa­

bilities and their

interconnections as well as on the mode, of opera­

tion and control. Some of the basic properties of the EMSs, related to the models below are (a) the possibility to work autonomously without any human interventi­ on; (b) automatic realization of basic and supplementary operations; (c) elimination of production line idle time; (d) maximum possibility for a complete preparation of a worKpiece on a single machine; (e) high economic efficiency; and others (cf e. g. [4]). The rapidly expanding area of FMSs requires new methods for mo­ delling,

simulation,

control and optimization,

as classical methods

are not always adequate. From the mid 80s, ( the earliest papers are probably [3-5]) the Petri nets (PNs) which have established themselves as a convenient me­ ans for FMS modelling. First of all, the FAB is a discrete system. Any modelling has to be based

on the concepts

An event corresponds to a state change. are associated with transitions.

of events

and activities.

when using Petri nets, events

Activities are

firing of transitions and/or marking of places.

associated with

the

Petri nets are appro­

priate for representing parallelism and synchronisation in an enginee­ ring environment. They are very useful at the global coordination

le­

vel where real-time decision making is required to execute the planned and scheduled production in the manufacturing shop.

The real-time de-

174-

CHAPTER 3

cisions of the coordination level also have to be consistent with manufacturing shop. of

The Petri net

rule-based system

where

the

can be seen as a control structure

the rules operate

on object

attributes

(parts, machines, tools) represented by tokens. The Petri net utiliza­ tion at the global coordination level increases the security and reli­ ability of the FMS Control. The application of Petri net theory to the FMS

is a very rich

domain. Let us now briefly sum up some advantages of this approach: (a) partial order among events

can be described

to meet

flexibility

requirements, (b) states and events are represented explicitly, (c) a unique family of tools is used throughout the specification, mo­ delling, qualitative validation and performance evaluation, imple­ mentation and the operational processes, (d) the same

family of

tools is used within

various functions of an

FMS and at various levels (scheduling, global coordination and lo­ cal control). It represents an important

aid for

integrating the

whole system, (e) an accurate,

formal description of the synchronization

mechanism

is provided, which is something essential to achieve security [7], (f) the graphic nature of Petri nets, with state graphs,

their conciseness in comparison

the possibility

of

analysing

them , and

the

existence of techniques for a direct implementation, (g) the

Petri net permits

direct realization of the modelled

system

from the analyzed net [8], One outstanding aspect ing

analysed to see

net

must satisfy

of Petri nets is their capacity for be­

if the constructed model is valid.

a set

of properties which are characteristic

"good model". Some of the properties are: tokens which any

For this the of a

boundedness - the number of

place in the net can accumulate during its evolution

is bounded; liveness - this does not only mean that the model is dead-

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

lock-free, exist

but

also that from each state reachable by the net

175

there

evolutions which can lead to the firing of any transition;

versability - if

re-

this is verified it means that it is always possible

(from any reachable state)

to return the system to its initial state.

For analyzing the properties of the Petri nets various techniques have been developed (a) Reachability

graph

analysis - this consists of analyzing the net

on the basis of the graph of the reachable marking from the initi­ al marking. This method allows a complete analysis of the net. (b) Reduction analysis - this analysis consists of applying to the net a set of rules which, while preserving the properties that are be analyzed,

gives a smaller model.

to

This method is complementary

to the previous method By first applying this method, later "rea­ chability analysis" is made easier. (c) Structural analysis - this method allows the analysis of many pro­ perties of Petri nets, independently of its initial marking. The main disadvantage of Petri nets arises from the size of the nets modelling very complex systems, such as FMSs. Another disadvantage is the fact functioning forces a

that a change in the system's

substantial change in the net structure.

Another suitable means for an FMS modelling are

coloured Petri

nets. They consist of places and transitions. A place can be marked by the elements

of a subset of colours

(none,

one or

various items of

each colour). The tokens of a place will be called coloured tokens

or

sinply, colours. The marking of a place can be represented by a formal sum of colours.

The resulting models can

be more concise than

obtained from Petri nets and the functions appearing

on the

those

net arcs

have a clear physical meaning. The main disadvantage of coloured Petri nets is the fact realization are

that techniques for model construction, analysis and not yet fully developed This is a result of this to-

176

CHAPTER 3

olJs relative newness. *

3. 6. 2

First

K

GN-model of fiMSs

In the FN-models of FAJSs the processes

constructed so far,

the dynamics

of

in FMSs is described and the data for main time parame­

ters of the modelled processes

are investigated (cf. [5]). Analogous

results can be obtained by GNs. On the other hand, the constructed ge­ neralized net models (QN-models) of FMSs can be used as tools for con­ trol of the processes, me.

which function sufficiently slowly in real-ti­

GN-models also give a possibility of optimization of the control­

led process

(we shall discuss this problem in

a further communicati­

on). Here we shall discuss a process, as described by PNs in [5]. The problem is to develop general understanding of which

the way

the performance of an FMS is affected by disturbances and

out how flexibility can attenuate these affects.

A GN,

in find

modelling the

same process is shown on Fig. 3. 8. It contains the transitions Z , Z , 1 2 Z , Z , having the following described components. Each of these tran3 4 sitions is activated on

every elementary

the functioning of the GN-model). arcs and of the places

The

are infinite.

time step

(this simplifies

capacities of the transition's The transition

Z

is activated 1

when there is at least one token in the places 1 , 1 , 1 and 1 and 1 3 12 13 simultaneously there is at least one token in place 1 . The transition 6 Z 2

is activated

the places 1 and 2

when there is at least one token in at least one 1 and simultaneously 14

of

there is at least one token

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

Fig. 3. fl

177

178

CHAPTER 3

in at least one of the places 1 and 1 5 9 The transitions Z

and Z 3

are activated -when there is at least 4

one token in at least one of their input places. The priorities of the places and transitions are 0 (i.e. they have no priorities). The tran­ sition conditions are 1 2

r

1

1

3

4

5

1 1

I W 1 1

W 2

W 3

false

1

I W 1 1

W 2

false

false

= 1

1

3

1 I 12 I

W 1

W 2

W 3

false

1 I 13 I

W

W 2

W 3

false

false

false

W 4

1 8

1

I false I

mtoere - "In place

W 1

1

there exists a token with

a characteristic which

8 corresponds to

the current identifier

of an available

machine

for the token (workpiece)"; W 2 W

= nW ; 1 = "In places 1 , 1 and 1

3

5 tic

8

there are no tokens with a cbaracteris-

9

which correspond to the current

identifier of an available

machine for the token (workpiece)"; - "There exists a workpiece which mist be machined";

W 4

SC&E APPLICATIONS OF GNs IN ECONOMICS,

1 6 6

1

1 II W W 22 || 55 rr s= 2

11 || 14 14 II

11 77

w w 7

INDUSTRY AND TRANSPORT

11 88

10 10

W W 6 6

ffalse alse

ffalse alse

false false

ffalse alse

ffalse alse

ffalse alse

false false

1 II ffalse alse 55 | |

ffalse alse

1 9

ffalse alse

W W

W W 66

W W 8 8

W W

W W 66

W W 88

55

II ffalse alse 11

11 99

55

179

where W

= "The machine which is machining the workpiece is at worK"; 5

= ~m ;

W 6

5

i. e. W

= "The machine breaks down"; 6

W

- "The machine

which is machining the workpiece

is at

work, or

7 there exists another machine

(a token in place 1 ), which can 9

machine the workpiece"; W

= "The machine is shelved"; 8 1 11 11 rr

= 3

11 II WW 66 11 99

1 12 12 WW 10

where W

= "The list of the processing machines is exhausted";

9 W 10

= nW ; 9 1 13 13 rr = 44

11 II WW 77 II 11 11

1 14 WW 12 12

where W

= "The repair duration takes a lot of time"; 11

180

W 12

CHAPTER 3

= 1W 11 In the transfer to the different places, the token

assumes the

following characteristics (the fact that the characteristic is not gi­ ven is reflected by the absence from the following list): $ -> "an identifier of the available machine "; 2 $ -> "the worKpiece cannot be machined"; 4 § -> "general time for processing of the token in the last token 5 transfer if the token was in place 1 ; general time for repa8 ir if the token was in place 1 "; 9 $ -> "general idle time, if the token was in place 1 "; 6 14 $

-> "the machine is not able to function"; 10

$

-> "general time for processing of the workpiece"; 11

$

-> "time for processing with the corresponding available machine". 12 The GN starts functioning at the time moment

T

according to a

* given time-scale and continues At the

t

elementary

time-steps.

start of the simulation process in place 1 there exist 8

tokens with the following characteristics; "identifier of processing machine; priority of the processing machi­ ne; capacity of the processing machine, etc. defined by

the GN-mo-

del' s user". At different time moments defined by a law (from the user), in the input place 1 tokens enter with the initial characteristics: 1 "identifier of the workpiece;

identifier's priority;

moment of its

arrival, a list of processing machines necessary for it; and others defined by the GN-model's user".

S(*E APPLICATIONS OF GNs IN ECONOMICS,

The token's

transfer in the GN

INDUSTRY AND TRANSPORT

181

depends on the truth-values of

the predicates of the conditions given above. As a result of the tran­ sfer, the tokens from both types receive, by the above functions, the following one of the GN-places,

characteristic

characteristics: idle time duration in every

general idle time

duration,

general time

for

processing in the GN etc. The tokens of the first type receive as cha­ racteristics the values of the general and mean times for the process­ ing of

the workpieces from the different types of machines

are more than 1), general and mean time for time

for the

functioning

repair,

types of machines

general and mean

of the machine between two repairs, etc.

The tokens from the second type receive values of the general

(if these

as characteristics the

and mean times for processing

in the different

(if they are more than 1), general and mean waiting

time for processing, general and mean time for repair of the process­ ing machine if a breakdown

occurs

during processing

user can define other characteristic function

time, etc.

values for

the

The

tokens

and as a result, corresponding new data can be received. When the real process is controlled by a GN-model in real time,

the characteristics

of the tokens are obtained from the real environment by the correspon­ ding interface. Ihe truth-values conditions

are calculated on the

of the predicates of basis of the ones

the transition

existing in

the

GN information, accumulated as tokens' characteristics, or as expert's (operator's) On

subjective

estimation,

the basis of a concrete

ficient following ons. cess.

information

or as

can be

accumulated

which

values

can be applied for

would show the future

ones

and thus

in real can

of

of both. time, suf­

be used

the random

GN-simulation

As a result, the GN-simulation processes

approximating the real

combination

GN-model functioning

definitions for changes of

These functions

a

of

in the functi­ the pro­

would be increasingly

the results of

the simulation

state of the process. In this case, the initial

characteristics of the tokens

ought

to be taken from the real

envi-

182

CHAPTER 3

ronment.

This information

will be necessary at different

management

levels. Further improvement of the GN-model of a FMS can be achieved by introducing the influence of the toolroom on its functioning. The con­ dition, flow and management of tools affect in a great extent the per­ formance of the system as far as a researcher wants to investigate mo­ re thoroughly the reasons for waiting because of a tool breakage, lack of tools, time delay for tool change, etc. In order to start machining,

a machine tool

tool set all the tools with enough tool-life.

must have

in the

Very often waiting for

machine tools is due to tool breakages resulting from uncorrectly cho­ sen modes of cutting or from defects in the in the machining of workpieces

are of a

blank parts. These pauses

great

significance

for the

characteristic of the system as a whole. GN-modelling the toolroom

and

allows one to introduce all these

attributes

tool-dependent conditions of the functioning

of

(i. e.

tool-life, tool breakage, availability of tools, time delay due to to­ ol change, etc. ) in order to achieve a more

realistic picture of the

behaviour of the FMS. M X

3. 6. 3 Second

OV-/axte7 of

K

fftSs

The constructed model is based on a real functioning the plant for metal-cutting machines in Sofia. The system (a) a store for workpieces, output operations zone

including a station for

the main

where loading/unloading is performed;

including fixing stations,

intermediate

system in

consists of

stations

input/ a fixing (ports),

and terminals; (b) machine park

including 12 processing machines of three

types depending

on the size

of the NC-palets,

washing

different machines

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

183

for clearing the workpieces, a measuring machine. To eliminate queues in front of the washing machines, are permitted.

If a measure

only two measure requests

request is not strictly

the palet with the workpiece waits on the cessing machine

determined,

rear buffer of the pro-

while at least one of the washing machines is re-

leased. (c) transport system including robocars in the machine area, in fixing zone,

robocars

robocars for transportation of tools between the

processing machines and the tool store. A model on Fig. 3. 9.

describing

the movement of

the workpieces

is

shown

It consists of four transitions with the following ele-

ments: (a) transition

Z with input places 1 , 1 and 1 and type v(l , 1 , 1 1 2 8 1 2

1 ) and output places 1 , 1 and 1 ; 8 2 3 4(b) transition Z with input places 1 and 1 and type v(l , 1 ) and 2 3 12 3 12 output places 1 , 1 , 1 , 1 . 5 6 7 8 (c) transition

Z 3

with input places 1 , 1 , 1 and 5 6 7

1,1,1 ) and output places 6 7 9

1 and type v (1 , 5 9

1 , 1 . 9 10

and output places
1 , 1 . 11 12

The capacities of the transition's arcs are 1, and those of the places are co with the exception of places 1 , 1 , 1 (which have capa5 6 7 cities equal to the numbers of the processing machines for small, middle and big workpieces)

and 1

with a capacity equal to 2 (there are

no more than two measure requests). Every transition is activated on every elementary In this model the transitions are

time-step.

activated when at least

one

184-

CHAPTER 3

Fig. 3. 9 of their input places contains at least one token. The transition conditions are 1

1

2 r i

= 1 1 1 2

IW 1 1 I W 11

1 IW A l l

1

3 W

4 false

2 W 2 W 2

W 3 W 3

where W

z "there exists a machine which can process the workpiece";

W 1

= nW ; 2

3

= "the list of the processing machines is exhausted"' '

W

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

11 55 rr = = 11 2 33 2

|| W W 11 44

11 II WW 12 I 4

12 I 4

11

11 66

11 77

8 8

W W 55

W W 6 6

false false

WW 5

WW 6

WW 7

5

6

185

7

where W = W v (-|W & W") v (nWJ & iV" 8, W ); 4 6 W = W" v (nW" & W ); 5 6 W = "the worKpiece is big"; 6 W

= "the workpiece is going for a further processing or 7 re";

and W

= "the workpiece is small";

W" = "the workpiece is a medium in size";

r

1 9

1

= 1

Il W 5 1 8

W

1

W

6

I W 1 8 I W 1 8

W

7

W

9

I W 1 8

3

1 1

10 9 9 9 9

where

W

= "the measuring machine is not free"; 8

W = iV ; 9 8 1

1 11

r

= 1 4

where

I W 10 I 10

12 W 11

to

the sto-

186

W

CHAPTER 3

= "the workpiece is shelved"; 10

w = -m . 11

10 During their movement

through the net,

the tokens

obtain new

characteristics by the characteristic function as follows: $ 3 $ 4

-> "waiting-time for place 1 ", 2 -> "general time for processing of the workpiece", "difference between the real time and the time which characteristic",

is given in

the initial

§ -> "name of the processing machine for small workpieces", 5 $ -> "name of the processing machine for medium-size workpieces", 6 $ -> "name of the processing machine for big workpieces", 7 $ -> "time for processing in 15 or 16 or 17, respectively", 9 $ 10 $

-> "waiting time in 1 ", 9 -> "value of the consuming labour",

11 $

->"time for processing separate steps". 12 Tbe places represent the following process components:

1 -> initial status of the process, 1 1 -> a workpiece is waiting in the store for processing, 2 1 -> the workpiece is going for processing, 3 1 -> the workpiece is going out of the system after processing, 4 1 -> the workpiece is processed on a machine for small workpieces, 5 1 -> the workpiece is processed on a machine for medium-size workpie6

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

187

ces, 1 -> the workpiece is processed on a machine for big workpieces, 7 1 -> the workpiece is going back to the store, 8 1 -> the workpiece is waiting for releasing of the washing machines, 9 1 -> the workpiece 10 ne,

is washed and measured from the measuring machi-

1 -> the size of the workpiece does not meet the requirements, 11 workpiece is shelved and going out of the system, 1 -> the workpiece is going for further processing, 12 is going to the store.

the

if necessary it

M *

§ 3. 6. ■*. Third

W-aode]

K

of FMSs

The system to be controlled is part of a flexible workshop the Renault company.

of

The purpose of the system is to mastic seal the

body work of automobiles. For this purpose it has six workstations (or simply stations),

each consisting of one work place P, sufficient for

one car body, and also one transfer bench with two roller tables,

one

designed to load the station (TL) and the other to unload it (TU). The bench, which is mobile,

can adopt two positions:

left,

to load, or

right to unload the car bodies. A system of transport, made up of six roller tables RT ) has 6

the task of transporting all

car bodies which

(RT 1 enter

the

system (via table RT ) to the sealing station, and of later unloading 1 and removing them to the exit via

RT6.

Each table

(TL, TU, RT ) can i

carry only one car body, and at any s t a t i o n there can be a maxirrum two (P-TU, P-TL, or TU-TL) at a given time [27],

of

188

CHAPTER 3

A coloured Petri net model delling the autonomous

coloured

of this system is Petri net

applied.

In mo­

model, we can distinguish

three stages: (a) independent modelling of different subsystems, which form

part of

the system under consideration, using a coloured Petri net. In the above-mentioned system, we can distinguish two subsystems:

trans­

port and stations; (b) fusion

of the modules by fusing

the transitions which

represent

synchronization between the subsystems. In the system, the loading and unloading of the stations requires synchronization between the transport and the station subsystems; (c) the eventual simplification of the coloured Petri net, obtained by eliminating implicit places. These are places whose marking can be expressed as a positive linear combination of

a set of

the net and whose elimination preserves the liveness

places of

and bounded-

ness of the net. With each place, we associate a set of colours such that we can distinguish all the tasks and resources associated with it. The urs can be simple or compound (n-tuples) depending on whether sociated

tasks or resources are defined

by one or

colo­ the as­

several parameter

With each transition, we associate a set of colours such that its car­ dinal is equal

to the number of different possible

evolutions of the

state of the system which are defined by their possible firings [27]. Below we

shall construct a GN which describes the above

pro­

cess modelled by a coloured Petri net. The graphical structure of this net is given on Fig 3. 10. The GN has no activated

local temporal

components.

Every transition is

when there is at least one token in its input places,

all transition types are of a disjunctive type. The capacities of the places are (cf. [27]):

i. e.

SCME APPLICATIONS OF GNs IN ECONCMICS, INDUSTRY AND TRANSPORT

189

Fig. 3. 10 c(l ) = c(l ) = oo, c(l ) = . . . = c(l ) = 6. 1 8 2 7 The priorities of the places are TI ( 1 ) = TT ( 1 ) = TI ( 1 ) >,

L

3

L

5

L

7

TI ( 1 ) = TI ( 1 ) = TI ( 1 ) = TI ( 1 > = TI ( 1 ) .

LI

L2

L4

L6

L8

The priorities of the transitions are equal. The capacities of the arcs are "a>". Every token enters the net at a certain time-moment (determi­ ned by a function 0 ) according to some fixed time-scale (with initial K o * time-moment T, elementary time-step t receive

an undetermined number

and time-duration

of characteristics

an initial characteristic "type of the car others)".

and

t ).

It can

enters with

body, current number

(and

190

CHAPTER 3

The GN contains 4 transitions and following

transition conditions

and to

8 places.

They contain the

the GN places, the following

characteristic functions are associated. 1 2 2 r 1

W

1 3 3

= 1 I W 1 1 1

W 2

1 I W 3| |1 3 1

w 2 2

= "there is an empty loaded table" & "there is an enpty roller tab1 le";

W 2

= iW ; 1

$ -> "
$

-> «, 3 1 4

W 3

1 5

r = 1 I W 2 1 3 2

W 4

1 I W 5 | | 3 5 3

W 4 4

a = "the workplace with a number pr x is enpty"; 1 cu

W = nw ; 4 3 $ -> " K; 5

SOME APPLICATIONS OF GNS IN ECONCMICS, INDUSTRY AND TRANSPORT

1 6 r

1 7

= l i I w 3 4 1 5

w 6

1 I W 7 1 16 7 6

W 7 7

5

a = "the unlaoded table with a number pr x is empty", 1 cu-l

e

= 1W , 5

W W

191

$ -> "«turation of the sealing of the car body, duration of waiting 6 in place 1 , duration of the transfer from the workplace to 7 the unloaded table">

$

-> »,

7 1 8 r = 1 | W 4 6 I 7 where W

; "there is an empty roller table" 7

§

-> "
of the transfer from unloaded table to

roller tab-

a le">. ■

» § 3. 6. 5 <3V-model of flexible

» manufacturing

cell

This model concerns the control system organisation

of

a fle­

xible manufacturing cell. We shall show how the Petri net model of the flexible manufacturing cell from [28] can be represented by GNs. A flexible

manufacturing cell

is used

tion of metal parts with specific dimensions. an

increasing

ganization

for

automatic produc­

It is

characterised by

automation of technological operations,

a better

or­

and a more accurate control finalization strategy aimed at

192

CHAPTER 3

increasing production.

The configuration of

a flexible manufacturing

cell, whose functioning is controlled by this model, contains: R 1

and R , machine tools 2

robots

(NC -milling machine, NC -lathe, NC -press), 1 2 3

two auxiliary buffers (B and B ), incoming T and outgoing T trans1 2 1 2 porters, a bin for rejected parts. The cell works in the following way: Raw pieces of material co­ me by the incoming transporter T

up to the take up position, which is 1

defined by a step movement. Robot R

waits in front of the transporter 1

T , grips the object and goes to the working space of a milling machi1 ne

NC

and there

leaves the workpiece.

1 back

and

waits for

After

processing

takes

the processed

a milling

After that, robot R 1

machine to

complete

the object, the robot goes to object

and

the processing.

the milling machine,

carries it over

in buffer B . Robot R takes the workpiece 1 2

to a free position

from the buffer

carries it over to the lathe. Then robot R

draws

draws back.

B 1

and

When the pro-

2 cessing on lathe NC is completed, R 2 2

takes the piece from

and transfers it to the press NC . Then, 3

the robot

B

the lathe

withdraws in 2

front of the press till the workpiece is pressed. After that the ro­ bot R takes the workpiece and carries it over to the transporter T . 2 2 Buffer B

is used for temporary

laying down

of the workpieces which

2 the lathe has finished

and waits to be pressed.

The reject

store is

used for storing the workpieces ruined during machining. A (3<-model, describing this process is shown in Fig. 3. 11, transition conditions in the form of matrix of predicates are

The

SOME APPLICATIONS OF CTfs IN ECONOMICS, INDUSTRY AND TRANSPORT

Fig. 3. 11.

193

194

CHAPTER 3

-- 1

r 1

I II

1

1 2

1

W 1

false

3

I false

W

2 I

1

where W 1

z "3(1 , TIME) = d(l >" 2 2

(the function

3(1)

gives the capacity of the place

1 and the func­

tion d(l, TIME) gives the nuntoer of tokens in the place 1); 1 4

1 5

r = 1 I W 2 2 1 2 1

F

IF

W

3 I

2

where W ; "3(1 , TIME) < d ( l )" & "the necessary time 2 2 2 p l a c e 1 i s run out". 2 1 6 1 44

r 3

I II

W 33

1

1 7

1 8

false

false

for

processing in

1 9 false

10 false

1 i false 7 1

w 3

nw & &W W 3 4

nW nW 8nw Snw & &w w 3 4 5

iw i w 8nw s n w 8nw 8nw &(W &(W vw » 3 4 5 6 7

= l1 |i f a l s e 1122 I

w W 3

iv nW & &W W 3 4

n nW W snw 8nw & &W w 3 4 5

nW -m ssnw n w snw snw &(W &(W vw ) 3 4 5 6 7

1

I false 14 14 I

W 3

nW n W& &W W 3 4

nW nW 8nW 8nW & &W W 3 4 5

nW nW 8nW 8nW 8nw 8nw &(W &(W vw ) 3 4 5 6 7

1

| false 17 I

W 3

-m & &W W 3 4

-m SnW 8nW & &W W 3 4 5

nW nW 8nW 8nW 8nw 8nw &(W &(W vW ) 3 4 5 6 7

where W = "d(l , TIME) < d ( l )", 3 6 6

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

W = "3(1 , TIME) < e(l )" & "the necessary time * 11 11

for processing

195

in

place 1 is run out", 6 W

= "d(l 5

W

, TIME) < 6(1 )", 13 13

= "the workpiece is good", 6

W

= "the workpiece is ruined". 7 1 11 11 rr = 11 || 66 | | 4

1 12 12

w w

false false 4

1 II ffalse alse 8 II

W W 4

1 13 13 r r 5

= 11 || 11 11 || 1 9

r r

W W 5

conclusion

W W 55

1 15 15

11 16 16

W W 66

W W 77

false false

false

W 7

1 I false 10 I In

false false

II ffalse alse II

:: 11 II 6 13 II 6 13

1 14 14

1 17 17

we shall note that the above constructed GN-mo-

del has important advantages in relation to the other Petri net models (a) it is considerably more universal; (b) as a result of

simulation

this model, more

of a flexible

manufacturing cell with

information on the process can be received (the

tokens

which enter the net

obtain

new

with certain initial

characteristics while moving

characteristics

through

the net by the

characterizing functions associated with the places); (c) the process (if it proceeds sufficiently slowly)

can be

control-

196

CHAPTER 3

led on the basis of the constructed model; (d) the process can be optimized on the basis of the model, K

K

3. 6. 6

Queue models Scheduling is

K

and GN-models in flexible

manufacturing

one of the important problems

in

the functio­

ning of the FMSs. The classical scheduling has

formulated

tactical

literature dating back to the 1960s,

scheduling

as

the following

problem: a job shop has n jobs and m machines. composed of a sequence of m operations,

optimization

Each job has a routing

a unique machine being assig­

ned to each. The goal is to find a schedule (the start time of im ope­ rations),

which minimizes some criteria,

e.g. makespan or tardiness.

A simple approximation of the computational complexity can be calcula­ ted as follows: a job has m operations, one on each machine, and since each machine can be scheduled in n! ways, m In!|

schedules.

it follows that

there are

A method for finding the optimal schedule would

be

the enumeration of all these schedules and choosing

the best one. The

search space, consisting of the possible number of

valid schedules is

very large. phase of

Traditionally,

scheduling has been

the planning process,

considered

the last

which gives a solution of the problem

as when to allocate a particular resource to an operation [59]. In practice this problem is solved in the following way: (a) the priority of the job is defined; (b) the jobs are ordered according to their priorities; (c) if the planned resource is not available,

the corresponding

jobs

with a lower priority must be transferred. In flexible manufacturing

the machines are ordered

way that they can meet the requirements

in such

a

of ful1 processing of a work-

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

197

piece in one place. The FMSs can be

considered

as one

capacity unit. Because of

centralizing according to objects, but not according to functions, the material flow can be

easily

observed.

In this way the transfer of a

job does not attach to the other capacity units. To solve the scheduling problems in the FMSs, some methods are used: (a) queue models; (b) methods of mathematical programming; (c) methods with priority rules, The GNs

are also used

for modelling

the functioning of

the

FMSs. We shall try to compare the queue models with the GN-models ac­ cording to the method of solving scheduling problems in FMSs. The queue model consists of

knots

workpieces in front of them The time

(machines) with a queue of

for processing

is random

The

workpieces are processed according to the order of their entrance into the system (first in, first out) or in a random way. After processing, the workpiece goes to another machine with some probability. cal

Analyti­

results are as follows: middle grade of loading up the machines,

middle quantity of workpieces in the system,

middle quantity

of wai­

ting workpieces [60]. One of the main disadvantages of the queue models for modelling FMSs

is the

value of

the analytical

results.

These results do not

match the real situation. It is due to several factors: (a) the often-used exponential distribution for the processing time is not close account

to the real distribution;

it is possible

deterministic processing time-moments only

cessing times are identical; (b) the processing sequence is strictly determined;

to take into when the pro­

i9fl

CHAPTER 3

(c> for the queue models an unlimited

capacity

is implied; this fact

does not match with the capacities of the machine buffer and

sto­

rage; (d) only one priority rule for scheduling

is used

(FIFO - first

in,

first out); (e) the workpieces

cannot be individually traced, the description be­

ing for the whole system; (f) the disturbances

(machine breakdown or lack of

tools)

cannot be

taken in account. The GN-model of FMSs removes the drawbacks of queue models: (a) we can use an arbitrary distribution of

the processing

times so

that in future extensions of the GNs one distribution can be ex­ changed with another; (b) the processing sequence is not strictly determined;

it depends on

the optimal choice at a particular time-moment; (c) the capacities correspond to the real system (machine buffer, sto­ rage ); (d) different priority rules can be used - rules for the optimal choi­ ce of the workpieces; (e) the workpieces can be individually traced; (f) the disturbances can be taken into account in the model. Comparing the de

two approaches of modelling FMSs, we can conclu­

that the GN-models

are more suitable

for deciding

the sheduling

problems in flexible manufacturing, K K

3. 6. 7:

X

Conclusion

In conclusion, note that

the above constructed GN-models have

the following important advantages in relation to other PN-models. 1) They are considerably more universal compared with

other PN-

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

199

models. A PN-model is constructed for a concrete type of FMSs. The GNmodel discussed here

describes the functioning of the results of

the

work of every element of a complete class of FMSs. 2) As a result of the simulation of more

information about the process

can

the FMS with this be extracted.

characteristic functions defined are constructed

in

GN-model

Note that the

such a way that

they are maximally similar to those for the PN-model. 3) The process

(if it proceeds sufficiently slowly) can be con­

trolled on the basis of the constructed model. 4-) The process of the

of the transfer of the workpieces and the choice

processing machines can be optimized on the basis of

the GN-

model. Ibis paper is based on [61-64].

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facturing system, Seminar,

Sofia,

Generalized net

First Sci.

representation of manu­

Session of

the Math.

Found.

Oct. 10, 1989, Preprint IM-MFAIS-7-89,

AI

Sofia,

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

207

1989, 60-62. [63] M

Stefanova-Pavlova, R. Christov, E. Dincheva,

net model of a flexible sion of the Math.

manufacturing

Found.

AI Seminar,

Generalized net

system, Second Sci. Ses­ Sofia,

March

30, 1990,

Preprint IM-MFAIS-1-90, Sofia, 1990, 40-42. [64] M

Stefanova-Pavlova, Queue models and Generalized net models

in flexible manufacturing. Found.

AI Seminar,

Third

Sci. Session of the Math.

Sofia, June 12, 1990, Preprint IM-MFAIS-

2-90, Sofia, 1990, 13-14.

208

CHAPTER 3

§ 3. 7: GENERALIZED HET MODELS OF THE ACTIVITY OF NEFTCCHIM PETROCHEMICAL COMBINE IN BOURGAS Stela Dimitrova, Ludmila Dimitrova, Trajana Kolarova, Plamen Fetkov, Krassimir Atanassov and Rumen Christov

Generalized Nets complex

objects

which

(GNs)

are a comfortable

means for designing

are characterized by a large scale of various

parallel processes taking place in the real time. In the general case, it is impossible to show analytical formulas formalizing these proces­ ses.

For their modelling,

various simulation methods are used. Petri

nets are one of these (new) basic methods. As discussed in

§1,1, the

GNs can also be used as a means for simulating such (real) processes, but (in contrast with the Petri tion condition predicates

nets and

by virtue

of their transi­

and characteristic functions) they can also

include in themselves elements of analytical methods for

description.

For example, some characteristic functions can assign to a given token a value that is calculated by analytical functions. "The NEFTOCHIH of the integrated strong

Petrochemical Combine (FCC)

industrial enterprises in

in Bourgas

is one

Bulgaria which exerts

a

influence upon the dynamic development of the State Industrial

Chemical Corporation, the economy of the country,

and the internatio­

nal division of labour. " [1] The processes going in PCC can be modelled by GNs and structed

models

can be used

for simulation

combine. The models are investigated following

the con­

of the processes in the the ideas

for hierar­

chical models and models based on the union and the composition of ot­ her simpler GN-models introduced in § 1. 1. The modelling starts with the construction of a simplified glo­ bal

model

of the combine in general.

It has the from of

Fig. 3. 12,

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

209

Fig. 3. 12 The GN used can be, e. g. , a second type intuitionistic fuzzy GN,

i. e.

a ON in which the tokens are replaced by "quantities" (see § 3 in App. 1) which

"flow in the net.

So a token ("quantity") enters place 1 1

and this

action

represents

ceived. This place symbolizes distribution

of

de: "chemical"

the quantity of raw materials (oil) re­ Petrochemical Sea Port (Terminal) where

raw material through the basic types of units is ma­ (1 ) and refinery.

The second one has the following

3 two components: a "Catalitic Reforming" (1 ) and an 4

"output of fuels"

(1 ). A part of the production from 1 is directed to 1 and 5 4 6

together

210

CHAPTER 3

with the production from 1 , is going (according to preliminary by set 3 conditions) to 1 9

(for production of plastics, fibres and rubber), or

to 1 (for production 10 A part of the 1 11

of petrochemicals - in "petrochemical units").

"quality" (token) is going from the last place to place

(for production of solvents),

and another

place 1 to the "catalytic cracKing unit" 12

part returns

through

and there takes part

with

newly entered "qualities", in the production of fuels (1 ) and 7 products (1 ). The toKens (qualities) 8 go to place

1 14

other

from places 1 , 1 , 1 , 1 , 1 9 11 7 a 5

and that symbolizes the outcome of the finished pro-

duct from the PCC. By this part of the GN the movement (in general outline) of fuels, petrochemicals and plastics, produced and processed in PCC can be observed. Besides, the GN has a second (not differentiated separately) part

that symbolizes

the entry of information on

the demand

of pe-

trochemicals (1 ), workout of this information (orders) (1 ) and re2 13 port on the orders (1 ) - the effect that they are satisfied or not. 15 The tokens ("quantities") that go into 1 have as initial cha1 racteristics the type of raw material, and during the time of movement in

the GN

they receive the following characteristics:

"the kind and

quantities of processing intermediate fractions" and "the kind and quantities of the finished product" depending on the type (Kind) of places they enter. obtain the

In the end in place

1 these tokens 14

("quantities")

finished characteristic "working expenses" and other

racteristics specified by the model's consumers.

cha-

SOME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

211

The other type of tokens are going to place 1 with the initial 2 characteristic "information about the demand of goods with sary description, quantities, with

the final

the neces­

parameters, etc. and are leaving the GN

characteristic "the order is satisfied" or "the order

is not satisfied" (as well as other data). More detailed processing

models can report on

the moment when the raw material enters the PCC, cesses,

such factors as

the duration of pro­

the moments when the products leave the PCC, the duration of

waiting for processing of raw materials or fractions etc. The other tokens can have their own priorities which determine the ways of

the transfer. The transition condition predicates can ha­

ve explicite

forms. During simulation of the PPC-processes, different

situations can occur in relation to the initial characteristics of the tokens in places 1 and 1 , and also in relation with the characteris1 2 tics

of the tokens which are in the net at

the initial

time-moment

(they correspond to the available quantities in the PPC). The described GN reflects only the most global connections exi­ sting

between the different

units of PPC. Mien more details

are re­

quired in the positions of the places and the transitions of the above net, new GNs will be

standing.

They will describe

the separate sub-

processes which are included in the most global processes. Another situation is the one when most global one)

corresponds

a new GN

(a subnet

of the

to a part of the places and transitions

of the above GN, but does not cover their content. A GN shown in Fig. 3. 13

is an example,

illustrating this. The

components of this net correspond to places 1 , 1 , 1 and the transi1 4 5 tion Z . 1 Briefly, the sense of the places of the new net are as follows: 1 - Petrochemical Sea Port (Terminal), 1

212

CHAPTER 3

Fig.

3. 13.

SOME APPLICATIONS OF GNs IN ECONCMICS, INDUSTRY AND TRANSPORT

213

1 - production of fuel oil 2 1 , 1 , 1 , 1 , 1 , 1 3 4 5 6 9

,1 - production of other cuts 13 30

1 - production of oil cut 7 1,1 - production of cut C 10 11 4 1 - production of high-octane gasoline 12 1 , 1 ,. . . , 1 16 17 25

- tanks

1 - production of cut C 26 5 1

- production of toluen.

27 The GN-models described here reflect the first steps modelling of

the NEFTOCHIM Petrochemical Combine

proposed text, other

of global

in Bourgas.

In the

aspects and methods for describing the petroche­

mical processes are not included. REFERENCE: [1] NEFTOCHIM Economic Combine-Bourgas, 1988.

BulgarReklama Agency,

Sofia,

214

CHAPTER 3

§3.8: A SBCCHD TYPE OF IHTUITIOHISTIC FUZZY GEHERALIZED HET-HCDEL IH THE CHEMICAL INDUSTRY Rumen Christov

and

Stoian Garbov

On the basis of the second type of intuitionistic fuzzy genera­ lized nets (IPGH2) (see § 3 in App. 1) we shall describe a part of the technological

scheme of the functioning of

a silo-farm with pneumo-

transport in a plasticware company. The raw materials are They are

ferried with

auto-cisterns

provided from 4 places and transferred to 10

and tanks. store-bunkers

(silos) or (if necessary) directly to the production lines. rials must flow continuously to these lines. duction lines from the silos

Raw mate­

The resin enters

6 pro­

by a distributor. Bach line has an indi­

vidual capacity and the material available for a definite time is con­ sumed. The IFGH2 describing this process has the form of Fig. 3.14. It has 3 transitions and 23 places with: - places 1 1

1 - symbolising the accepting places; 4

- places 1 ,..., 1 - symbolising the silos; 5 14 - place 1

- showing the transportation of the material

to

the pro-

15 duction lines; - place 1 - modeling the distributor; 17 - places 1 1 - symbolising the flow production lines. 18 23 All transitions have equal priority. The places 1 and 1 15 ve

the highest priority;

the other places have equal priority.

transition conditions have the form:

ha-

17 The

SC*E APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

Fig. 3. 14

215

216

CHAPTER 3

1

1 5

1

=

1 7

1

1

8

1 10

9

1 11

1 12

1 13

1 14

15

I 1 1

W

W 2

W 3

W 4

W 5

W 6

W 7

W 8

W 9

W

W 10

11

W 1

W

W 3

W 4

W 5

W 6

W 7

W 8

W 9

W

W 10

11

1 3

I | I I 1

W 10

11

1 4

I 1

W 10

11

W 10

11

1 r

1 6

1 2

1

1

I 16 I

2

W 1

W 2

W 3

W 4

W 5

W 6

W 7

W 8

W 9

W

W 1

W 2

W 3

W 4

W 5

W 6

W 7

W 8

W 9

W

W 2

W 3

W 4

W

W

W 1

5

W 6

W 7

W 8

W 9

where W = "the r e s i n m i s t go t o t h e i - t h s i l o " i - "the resin nust go to

W

(1 i i i

10),

the flow production lines,

because there

11 exists a flow

production line with resin less than the critical

quantity"; 1

1 5

r 2

=

I 1

W

1 I 6 |

W

1 7

I 1

W

1 8

I 1

W

1

I

W

9

16

1 17

12

W 13

12

W 13

12

W 13

12

W 13

12

W 13

12

W 13

12

W 13

12

W 13

12

W 13

12

W 13

1 1

I 10 I

W

I 11 I

W

I 12 I

W

1 I 13 I

W

1

W

1 1

I 14 |

SCME APPLICATIONS OF GNs IN ECONOMICS, INDUSTRY AND TRANSPORT

217

where W

; "the resin must go to another silo"; 12

W

= "the resin must go to

a production

line with

resin less than

13 the critical quantity or with a minimal resin quantity"; 1

r

= 3

IS

1 19

1

1 20

1 I W W W 15 I 14 15 16 I 1 I W W W 17 I 14 15 16

1 21 W

1 22 W

17 W

18

W 19

18

W 19

W 17

23

where W

= "the resin mist go to the i-th production line which has

the lo-

i west resin quantity"

and

"the resin is below the critical li­

mit" (14 i i i 19). The characteristic functions $ give the infinvm and the suprei mum of the available resin quantity in the place 1 (1 i i £ 23). i The IPGN2-models functioning starts at a certain time-moment according to a certain absolute time-scale o step (t ) being 2 minutes.

and the elementary

T

time-

The features of this net are the discrete values of the predi­ cates of the transition the tokens are

conditions but the net

is an IFGN2,

"quantities" which flow in the net and

part

because of these

quantities is lost in the process of transportation, i. e. there exists a degree of indetermination. The following problems are solved (addaptively) by the model: a) The optimal schedule of the raw materials ferrying is determined. b) TTie optimal loading of the machines is determined. c) The optimal capacity of the connected elements of the manufacturing is checked.

218

CHAPTER 3

d) The possible waiting of the machines are determined. On the basis of the IFGN2-model, the processes in a large class of productions of the chemical industry can be simulated and led (in real-time).

control­

Chapter

4:

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

The first research in the area of medical

diagnosis dates back to 1983.

structed nets are collected. scribed models

by means of

generalized net modelling of In § 4. 1 almost all

The authors plan to realize all program package for

the de­

generalized nets.

more general model of diagnostics is introduced in § 4. 2.

219

the con­

A

220

CHAPTER 4

H I :

APPLICATIONS OF GENERALIZED NETS IN HEFHROLOGY

Joseph G. Sorsich

and

Krassimir T. Atanassov

In this chapter, an exanple of the application of

GNs in medi­

cine is given, Nephrology (see e.g. [1-16]), as a relatively new field in medicine,

has been used to make things clearer

and to demonstrate

the possibilities of GNs. Some other applications in medicine are also available (e.g. [17]). Hie mathematical

foundations of

our research

are essentially

different from other basic mathematical methods applicable in medicine (see e. g. [18-21]). A renal

damage is usually suspected in the course of a routine

examination of a patient with some clinical signs and symptoms ria, renal colic,

edema,

turbid urine),

some disturbances in blood sample or urinalyse are present themia, proteinuria, etc. ). kidney deseases,

(byperazo-

Based upon these common signs relating to

the following GN is constructed. Ibis is not a defi­

nite model and it may be a subject of further modifications. which will

be

(dysu-

arterial hypertension or if

New GNs,

the subnets of the general GN described below,

added, and/or the subnets themselves may be modified. to demonstrate that such a subject in biological

may be

Our aim is also

sciences as medicine

is susceptible to comparatively simple and easy to understand mathema­ tical formulation by GNs. The same may be realized in almost all other areas of medicine. Initially, we shall describe one as subnets other GNs

(general)

describing the process

signs and symptoms in nephrology.

GN

which contains

of diagnosing

Let this GN be E.

different

It has one input

place L . The tokens of this GN will represent the patients of a nephrologic unit. Everywhere we describe the transfer of one token the tokens generated by it,

(patient)

or

but this is valid for every set of tokens

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

221

(patients) transferring in the GN. At the end of this chapter we shall discuss these problems more widely, as well as

the possibilities

for

application of this and similar GNs in different medical units. The token goes to place

L

with

the initial

characteristics

1 "patient with probable Kidney disease", kens which

represent the appropriate

next it splits into investigations of

four to­

the patient.

Later these tokens may be united into one token whose characteristics correspond to all the applied investigations, Part of these tokens may leave the GN if some of the signs are missing. The sense of the diffe­ rent GN's places are as follows: L - abnormal findings (changes) in urine are present, 2 L

- elevated blood urea nitrogen (BON) is present, 3

L - clinical signs and symptoms of renal disease are present, 4L

- arterial hypertension is present, 5

L

- permanent proteinuria is present, 6

L

- hematuria is present, 7

L

- turbid urine is present,

a L

- renal colic is present, 9

L

- dysuria is present, 10

L

- edematous patient is present. (Edemas of renal origin are always 11 related to proteinuria, refer to § 9. 1),

L

- evaluation of arterial hypertension, 12

L

- a specifying the type of renal damage, 13

L

- arterial hypertension of renal (renoparenchimal) 14

origin is sus-

222

CHAPTER 4

pec ted, L

- renovascular hypertension is suspected. 15 The transition conditions are as follows: L 22 rr

= L 11

L

ll ttrue rue

4 4

ttrue rue

true true ;;

11 I I L

L 66

rr

L 33

8 8

V V

= = L

II W W 22 11 11

2

L 77 W W

22

3

where W

= "permanent proteinuria is present", 1

W

= "hematuria is present", 2

W

= "turbid urine is present"; 3 L 99 rr

L

L 10 10

= L II W W 44 11 44 3

11 11

W W 55

W W 6

where W

= "renal colic is present", 4

W

= "dysuria is present", 5

W = "edematous p a t i e n t i s present"; 6 L r

= L 4

L 12

13

l true

true

5 I L

L 12

r

= L 5

where

I 113 3 I

W 7

13

W

fl8

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

223

224

W

CHAPTER 4

= "there is suspicion of renoparenchimal arterial hypertension", 7

W

= "there is suspicion of renovascular arterial hypertension". 8 From the fact that a token

(a unique one or one of many repre­

sentations of a given token (patient) that enters GN

E) enters places

L , L , L , . . , , L , L , L , i t follows that this token is directed 3 6 7 42 14 15 to the following subnets: L

- to the GN from § 4. 1. 1, 6

L

- to the GN from § 4. 1. 2, 7

L

- to the GN from § 4. 1. 3, 8

L

- to the GN from § 4. 1. 4, 3

L

- to the GN from § 4. 1. 5, 9

L

- to the GN from § 4. 1. 6, 10

L

- to the GN from § 4. 1, 1, 11

L

- to the GN from § 4. 1. 7, 12

L

- to the GN from § 4. 1. 8, 14

L

- to the GN from § 4. 1. 9. 15 On the basis of this model, we can achieve the following:

- control - optimization -simulation of real processes. Control initial

is realized

state of the patient

the token in L ). 1

as follows:

the physician describes

(this is the initial characteristic

the of

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

225

In the other offices, the token (patient) gets as current characteris­ tic the

results of the corresponding investigation. Thus on the basis

of the total

(at the moment)

the basis of all existing

information about the patient

(i.e. on

characteristics of the tokens, generated by

the initial token (patient)),

a decision for the subsequent direction

of the tokens is determined (i.e. the subsequent patient's

investiga­

tions). When the transfer of more tokens (patients) we can optimize

in the net occurs,

their transfer with the aim of achieving minimum wai­

ting-time in queues. Knowing the probable distributions

connected with

the tranfer

of the patients in different offices, on the basis of the total GN-model, the process can be simulated. pital units can be determined

Thus, different parameters of hos­

(e. g. workload of offices, average time

for servicing a single patient and of stay of a patient in a hospital

a single office,

unit etc. ).

average time-

This information

can be

used for optimal organization of different medical units. Note, that direct use

of our model is possible

(in this form)

in specialized (nephrological) units because only the processes of the nephrological disease diagnosis are described in it. If we want to ap­ ply similar models for other medical purposes

(cardiology, neurology,

obstetrics, laboratory investigations and their interpretation, etc. >, we must include in it GN-models of the corresponding case. Some Bulga­ rian physicians are working on this problem There already exist more expert and/or informative medical sys­ tems to help physicians in diagnosis and therapeutics. Such expert and informative systems may be included in the framework of the described total GN-model or in some of its parts (subnets).

For example, expert

and/or informative systems for diagnostics can be used frames of the global GN's memory

(see App. 1)

(e. g. , in the

for calculation of the

226

CHAPTER 4

truth-values of some

transition condition predicates and for calcula-

tion of some toKen's characteristics. On the other hand, expert and informative systems for therapeutics may be included in the described GN-model, when the final token's characteristics are to be determined. Below we describe other GN-models of and symptoms

basic nephrological signs

but these models have an independent sense because their

entire inclusion in the GN

E deprives them of

the clearness which is

easyly attainable by the GN-tools. This paper is based on [22-30],

. § 4. 1. i Permanent

»

«

*

proteinwla

Any permanent proteinuria in adults is

a pathological situati-

on. Proteinuria may be detected in different conditions. Usually it is asymptomatic, but the practitioner's first step must always be a careful physical examination of the patient - not only

the urinary tract,

but the cardiovascular system (including arterial blood pressure, eyeground) and other systems as well. The tokens (patients) get into place racteristic

"permanent proteinuria".

1 l

Next they

with the initial chasplit and get in the

sections with initial places 1 , 1 and 1 2 3 4-

so that each gets

propriate

the complete

characteristic:

"the result of

the ap-

urinalysis",

"plasma creatinine's level (or gromerular filtration rate)" and "X-ray examinations of the urinary tract". In the last case, the token leaves this GN and gets in the GN and r

2

have the forms:

from § 4. 1. 2. The transition conditions r 1

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

r 1

= 1 | 1 I

1 2

1 3

true

true

1 5 rr 22

= = 11 II 22 II

The token in place

1

1 2

1 4 true

1

1 7

6

ttrue rue

ttrue rue

227

ttrue rue

8 true true

splits into four tokens which get

into

places 1 , 1 , 1 and 1 according to their characteristics: amount of 5 6 7 8 24-hours proteinuria, presence of glucosuria, microscopic investigati­ on of urine sediment, urocultures. The transition condition

rr 33

11 99

11

- 11 II W W 33 || 11

W W

10 10 2 2

where W

= "finds out renal insufficiency", 1

W 2

= ~\V . i

The token leaves this GN also and gets into the GN from § 4. 1. 4. 11 11 11 rr = = 11 II W W 44 55 | | 33

11 12 12 W W 4 4

where - "proteinuria more that 3 gr, per 24 hours",

W 3

w - -w . 4

3 11 113 3 rr 55

where

= = 11 II W W 66 II 55

11 14 14 W W

6

228

CHAPTER 4

W = "there is glucosuria" 5 (the token leaves this GN - the patient is directed to diabetologist>,

w =

5

6

r = 1 1 6 7I The token in place places 1 , 1 15 16 results

of

and 1 17

1 7

1 15

1 16

true

true

1 17 true

splits into three tokens which get into

with the corresponding characteristics:

the microscopic examination of

urine sediment

the

for white

blood cells, red blood cells, bacteria The transition conditions r , r 7 8 1 18 r

7

- 1 I W 8 1 7

r

have the forms 13

1 19 W 8

wher W = "urine cultures are positive" 7 (the token leaves

this GN and

the patient is

directed toward the GN

described in § 4. 1. 3), W = -W ; 8 7 1 20 r = 1 8 H

| W I 9

1 21 W

10

where W = "there is a paraproteinuria" 9 (the token leaves the GN - the patient is directed to hematologist), W

10

: nW ; 9

APPLICATIONS OF GENERALIZED NETS IN JffiDICINE

229

230

CHAPTER 4

1 22

1

r = 1 I W 9 12 I 11

W

23 12

where W

= "the proteinuria is combined with hematuria" 11

(the token goes to the GN from § 4. 1. 2); W 12

= nW ; 11 1 24 24 r 10 10

= 1 I W 15 15 I I 13 13

1 25 25 W 14 14

where W

= "leucocyturia is present"; 13

W 14

= nW ; 13 1 26 26 r 11 11

= 1 I W 16 I 15 16 I 15

1 27 27 W 16 16

where W

= "hematuria is present"; 15 = IV

W

16

15

r 12 12

1 28

1 29

= 1 | W 17 17 I I 17 17

W 18 18

where W

= "bacteriuria is present"; 17

W 18

= nW ; 17

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

231

1 30

r

= 13

If the token

1 I 21 I

true

1 I 23 I

true

1 I 14 I I| 1 I 25 I

true true

1 I 27 I

true

1 I 29 I

true

1 I 19 I

true

1 I 10 |

true

gets into place 1 or 1 , it enters the GN from 2428

§ 4. 1. 3 and from place 1 26

to the GN from § 4. I. 2.

Acute nephritic syndrome takes shape if 1 , 26

in places

1 , 1 5 9

the tokens arising from an initial one are gathering

and

(often the

hematuria is macroscopic, the renal insufficiency is transitory, there are edema and arterial hypertension).

The tokens

in places

1 , 1 , 10 14

1 ,1 ,1 ,1 ,1 and 1 transit in place 1 - the patient 19 21 23 25 27 29 30 directed to a nephrologist,

is

where complementary investigations are to

be undertaken (umunological, renal biopsy, etc. ). K K

»

§ 4. i. 2 Hematuria The

presence of blood in urine (hematuria) is a conmon symptom

in nephro-urology. The causal diagnosis sometimes is very complicated.

232

CHAPTER 4

Hematuria may

be isolated

or combined with other signs and symptoms.

Thus a complete examination

of the patient is always indispensable.

The token gets into place 1 with the initial

characteristic -

1 presence of hematuria,

which is confirmed by a microscopic investiga­

tion of the urine sediment. The transition condition 1 2 r

= 1 1

1 2

I W

W

1 1 1

2

where W

= "there is a macroscopic (gross) hematuria"; 1

W 2

= nW . 1 The token in place 1 2

splits and gets into places

with the appropriate characteristics:

results of the test

glases and X-ray examinations of the urinary tract,

1 4

and

1 5

with three

because the tran­

sition condition is 1 4 r = 1 | true 2 2I Place 1 6 stration

1 5

1 6

true

true .

is included in the framework of this

GN

as an illu-

of one of the methods for constructing a general GN.

presents the relation between the GN from § 4. 1. 1 and

It re­

the net presen­

ted here. In the framework of the general GN, tokens will transfer he­ re from place 1 of the GN from § 4. 1. 1 and this corresponds to trans4 ferring from one net to another. 1 9 9 r = 1 | I W 4 4 1 3 4 4 1 3

1 10 10

1

W

W w 4 4

11 11 5 5

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

233

234-

CHAPTER 4

where W

= "the presence of initial hematuria"; 3

W

= "the presence of terminal hematuria"; 4 - "the presence of total hematuria".

W 5

When the token gets into places 1 9 directed to a urologist

r

= 5

(1

and/or

1

the patient is 10

). 19 1 12 12

1

1 5

I true 1

W

1 6

I true 1

W

1

13 13 6

14 14 W 7 W

6

7

1 I true 7 |

W 6

W 7

1 l true 15 I

W 6

W 7

where W

= "there are no contraindications for intravenous urography"; 6

W 7

= ~W . 6

Thus this examination is performed, Transition conditions

r , r , r , r ,r ,r ,r 7 a 9 10 12 14 15

have the forms 1 19 r

= 7

1 | true ; 9 I 1 I true 10 I

and

r 16

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

11 21 21

235

11 20 20

11 II W W 12 II 99 12

W W

rr = - 11 II W W 88 113 3 II 99

W W

11 II W W 114 4 II 99

W W

10 10 10 10

10 10

where W

= "there are

no urographically

detectable changes or

the results

9 are uncertain"; = IV

W

10

9

(in this case the patient is directed to a urologist); 11 222 2

11 23 23

rr = = 11 II W W 99 220 0 II 11 11

W W 12 12

where - "there are no cystoscopic changes

W

or the diagnosis

is uncerta-

11 in";

. -m ;

w 12

11

rr 110 0

11 224 4

11 25 25

11 26 26

= = 11 II W W 222 2 II 113 3

W W 114 4

W W

11 II W W 13 333 3 II 13

W W 1144

W W

15 15 15 15

where W

& IV

= 1W 13

14

15

(the token gets

into place 1 - the patient 24

is left under

lance); W = " t h e r e i s a n e e d f o r complementary X-ray 14 W = "renal biopsy i s 15

needed"

examinations";

surveil-

236

CHAPTER 4

(the token in place 1 usually gets 26 agnosis); 1 29 29 r

= 1 I W 25 I 16 25 I 16

12 12

as characteristic a definite di-

1 30 30

1 31 31

W

W 18 18

17 17

where - "an extended examination of the

W

renal parenchyma is needed";

16 W

= "an examination of the bladder is needed"; 17

W

= "an extended examination of the

vessels is needed" ;

18

r 14 14

1 34 34

1 35 35

1 36 36

= 1 I W 29 19 29 I I 19

W 20 20

W 21 21

where W

= "an arteriography is required"; 19

W

= "an sonography is required"; 20

W

= "a scanning is required"; 21

r 15 15

1 37 37

1 38 38

1 39 39

= 1 I W 30 I 22 30 I 22

W 23 23

W 24 24

where W

= "a retrograde cystoscopy is necessary"; 22

W

= "a retrograde ureteropyelography is necessary"; 23

W

= "an anterograde pyelography is necessary"; 24

r 16

1 40

1 41

= 1 | W 31 I 25

W 26

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

237

where W

= "an investigation of the renal arterial system is indispensable" 25

(aortography, selective renovasography), - "an investigation of the renal venous system is indispensable"

W 26

(cavography, renal phiebography). The section of the GN described above ends by a transition with condition 1 42

1

| 11 I

W

W

1 I 34 I

W

1

I 35 I

W

I 36 I I| 1 | 37 I

W

1

1

r

= 17

1

W

( 39 I

W

1 I 40 I

W

1 41

W

1

I

27

28

27

W 28

27

W 28

27

W 28

W

I 38 |

43

W 27

28

27

W 28 W

27

28 W

27

28 W

27

28

where W

= "urologic cause of hematuria is found" 27

(the patient is directed to a urologist - 1 ); 44 W

= "a nephrologic cause of a hematuria is found". 28

(the patient is directed to nephrologist - 1 ). 45

238

CHAPTER 4

Transition conditions r and r have the forms 9 13 1 27 r = 1 I W 9 23 I 29

1 28 W 30

where W

= "bleeding from the lower urinary tract is discovered" 29

(and the token gets into place 1 ); 44 W

= "bleeding from the upper urinary tract is discovered"; 30 1 32 r

1 33

= 1 I W 13 28 I 31

W 32

where W

= "the bleeding is unilateral", 31

(and the token gets into place 1 ), 44

w

= nw

32

31

(the toKen gets into place 1 and the condition r is verified). 33 10 Transition conditions r 3

and r have the forms 6 1 7 7

r 3

1 8 8

= 1 | W 3 I 33

W 34

where W

= "presence of hematuria combined with proteinuria

that cannot be

33 explained by the presence of blood in urine only"; - "presence of isolated hematuria";

W 34

1 1 1 15 16 r = 1 | W 6 8 I 35

W

1 17 W

36

18 W

37

38

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

239

where - "X-ray examination is needed"

W 35

(and a token's transfer to a 5-th transition follows); W

= "there are definite signs of nephropathy" 36

(the patient is directed to a nephrologist - 1 ); 45

W = "investigations of the concomitant proteinuria i s needed" 37 (the token leaves this GN and gets in a GN from § 4. 1. 1), W

= "investigation of the concomitant arterial hypertension is need38 ed"

(the token leaves this GN and gets into place L

of the basic GN E and 5

from there it goes to some of the GNs from § 4. 1. 7, § 4. 1. 8 or

§ 4. 1.

9). K K

K

§ 4. I. 3 Turbid urine Here some advantages of GNs and their applications

in the ini­

tial steps of identification and investigation of a turbid

urine sam­

ple will be illustrated Tokens (patients) enter place 1

with the initial characteris-

1 tic

"complains of

voiding turbid urine".

The tokens'

place 1* to place 1" shall be denoted by V -> 1". The transition condition of the GN are 1 12 2 r 1 I true r - 1 I true ; ; 1 1 1 1I I

transfer from

240

CHAPTER 4

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

1

1 2

r

= 1 2

3

I W 2 1 1

W 2

where

W

= "acidification of the urine specimen is performed"; 1

W

= "investigation of urine with test strip"; 2 11 55 rr 33

11 6

= = 11 II W W 33 11 33

W W 4 4

where W

= "the turbidity of urine disapears"; 3

W 4

= IV ; 3 11 77 rr = = 11 4 4

4

11 8

II W W 1 5

W W 6

4 1 5

6

where W = "there a r e no changes i n t h e t e s t 5

strip";

w = -m ; 6

5 1

1 9

1 5

I I

W 7

1 r

= 5

26 W 8

| true 6 1 I 1 I W 7 | 9

false

1

false

| true 8 I

W 10

where W - "the t r a n s f e r 7

1

-> 1 4

i s a c c o m p l i s h e d "; 8

24-1

242

W 8 W

CHAPTER 4

= nW ; 7 = "the transfer 1

9

3

-> 1 is accomplished "; 6

- -\V ;

W

10

9 1 lO r 6

1 11

1 12

= 1 I W W W 9 I 11 12 13

where - "bacteriological investigation of urine is undertaken";

W 11 W

= "microscopic

urine sediment

examination is carried out";

12 - "macroscopic examination of urine is accomplished";

W 13

1 1 13 14 13 14 r = 1 I W W 7 10 14 15 7 10 I I 14 15 where W

= "there is bacteriuria"; 14 - iW

W

15

;

14 1 15

1 16

r = 1 I W S 8 11 I 16

W 17

where - "there is leucocyturia";

W 16 W 17

= iW ; 16 1 17

1 18

r = 1 I W W 9 12 18 19 9 12 I I 18 19 where

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

- "there are urethral filaments in the urine specimen",

W 18

- -W ;

W 19

IS 1 9

1 26

1 r

= 10

I W false 15 I 20 I| 1 I false W 16 I 21

1

I W 17 I 20 17 I 20

W 21 21

where W

= "the transfer 1 20

W

-> 1 11

= nW 21

is accomplished"; 15

; 20 1 9 r

= 11

1 I true 20 I

;

1 I true 21 I

r 12

1 25

1

= 1 I W 24 I 22

W

31 23

where W

= "there is no prostatic hypertrophy"; 22

W

= iW 23

; 22 1 19 1

| true 13 I r = I 13 1 | true 14 I 1 I true 25 I

;

243

244

CHAPTER 4

1 27 r

= 14

1 I true 26 I ; 1 | true IS I 1 22

r 15 15

1 23

= 1 I W W 19 19 I I 2424- 25 25

where W

= "there

are no

X-ray

alternations

in

the kidney

and urinary

24 tract"; W

= nW ; 25 24

r 16 16

1 ] 29

1 30

= 1 I W 23 23 I I 26 26

W 27 27

where W

= "X-ray alternations in the urinary tract are demonstrated" 26 & -W

;

27 W

= "X-ray alternations in the Kidney are present". 27 The transfer 1 -> 1 means that it is indispensable to inves1 2

tigate a fresh urine specimen.

The transfer 1 -> 1 means that the 5 26

turbidity of urine disapears and thus the investigation is discontinu­ ed. The transfers 1 - > 1 , 1 - > 1 and 1 -> 1 impose an extension 5 9 7 9 8 9 of investigations. The transfers 1 15

-> 1 20

and 1 17

taneously performed, which means that leucocyturia ments are found. The transfers also

-> 1

are simil-

20 and urethral fila­

1 -> 1 and 1 -> 1 , which are 16 21 17 21 simultaneously performed, signify the absent leucocyturia, but

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

245

there are urethral filaments. The token in place 1 takes ascharacte24 ristic the result of the investigation of the prostatic gland, place

1 , the characteristics 19

of the results from

and in

the intravenouse

urography. The transfers 1 -> 1 and 1 -> 1 mean the end of the 26 27 18 27 investigation; 1 -> 1 means that the patient is directed to a spe23 29 cialised nepbrologic unit; 1 -> 1 30 32 patient is

and

1 -> 1 means that the 31 32

directed to urological unit; 1 -> 1 means that the pa22 28

tient remains under observation after appropriate therapy.

• § 4. 1.4 Elevated blood urea Estimation of importance

»

nitrogen

the Glomerular Filtration Rate (GFR) is of great

both in the initial evaluation and in subsequent

low-up of patients with renal disease.

the fol­

Measurement of blood urea nit­

rogen (BUN} is probably the most commonly used, albeit the least accu­ rate, clinical method for estimating GFR. The serum creatinine concen­ tration provides a more reliable estimate of GFR. In place 1 we have a token (patient) with the initial charac1 teristic

"elevated BON". Next it splits and gets in places 1 , 1 and 2 3

1 , as the transition condition r has the form 4 1 1 2 - 1

r 1 Each

place

namics

I true 1 I

1 3 true

1 4 true

gets an appropriate characteristics

investigation"

(1 ); "state 2

"results of barody­

of bydratation - evaluation

of

24-6

CHAPTER 4

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

247

extracellular volume" (1 ), and "urinalysis" (1 ). 3 4 The transition condition 1 5 5 r 2

1 6 6

= 1 I W 2 1 1

W 2

where W

= "there is a reduction in effective arterial blood volume"; 1

W 2

= IV . 1 The t r a n s i t i o n c o n d i t i o n

r r

1 7

11 88

= 11 II W W 3 33 11 33

W W 44

1 9 W W , 5

where W

= "there are symptoms of hyperhydratation"; 3

W = "normal findings"; 4 W

= "dehydration is present". 5 The transition condition

r 4 4

1 10

1 11

1

= 1 I true 4 4 I I

true

true

1 I true 18 |

true

true

12

From place 1 the token splits and gets into places 1 (deter4 10 mination of urea in the urine), protein), 1 (examination 12

1 11

(evaluation for

the presence of

of urine sediment for broad waxy casts).

The transition conditions

r , 5

r

and 6

r

have the following 7

248

CHAPTER 4

forms: 1 13 13 r

= 1 I W 5 10 I 6

1 14 14 W 7

where W

- "there is an increase of urea excretion in urine"; 6

W

= "there is a decrease of urine urea" ; 7 1 15 15 r = 1 I W 11 I 8 6

1 16 16 W 9

where - "proteinuria is present" (see § 4. 1. 3);

W

a w = nw . 9

8 1 17 17

1 ia ia

r = 1 I W 7 12 10 7 12 I I 10

W 11 11

where W

= "broad casts are present"; lO = lW

W

11

10

and this investigation may be repeated (see § 4. 1. 3). Ihe tokens from

places 1 , 1 and 1 5

9

get into place 1 13

(the-

19

re are signs of functional, reversible renal failure). The transition conditions

r

and r 8

have the following forms; 9

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

249

1 19 1 6 r

I I I | I

= 8

1 9 1

I 13 I

true true true

1 20 r

= 1 9

I W 19 I 12

1 21 W 13

where W

= "BUN is normalized after an appropriate therapy"; 12

.

= IV

W 13

12 The tokens from places 1 , 1 , 1 , 1 ,1 ,1 6 7 8 1415 17

and

1

get 21

into place 1 because organic renal failure is suspected. 22 The transition conditions r

and r lO

have the following forms: 11

1 22 1 5

r

I | I

true

1 I 21 I

true

1 7

true

= 10

1 8 1

l I I I I

true

| 14 I

true

1 | 15 I

true

1

true

| 17 I

250

CHAPTER 4

1 23 r

1 1 2* 24 25

= 1 I true 22 I

11

true

true

Tokens from place 1 mist be submitted to further investigati22 ons: blood (1 ), ultrasonographic (1 ), and radiologic (1 ) evalua23 24 25 tions of kidneys. The transition conditions r

, r , r and r have the follo12 13 14 15

wing forms'.

r 12

1 26 26

1 27 27

= 1 I true 23 I

true

1 1 1 32 33 32 33 r 13 13

= 1 I W 24 I 14 24 I 14

W

1 I W 25 I 14 25 I 14

W

1 34 34

35 35

16 16

W 17 17

16 16

W 17 17

W 15 15 W 15 15

where W

= "the size of both kidneys is normal"; 14 - "presence of bilateral small kidneys";

W 15 W

= "the kidneys are enlarged"; 16

W

= "there is enlargement of the pyelocalicial system"; 17

and

urologist must be consulted because obstructive

pected;

r 14 where W

= "anaemia is present"; 18

1 28

1 29

= 1 I W 26 I 18 Id

W 19

uropathy is sus­

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

w

- -m

19

251

;

18 1 30 r 15

1 31

= 1 I W 20 27 I

W 21

where W

= "there is hypocalcemia"; 20

w

- -m ;

21

20 1 36 1 I 28 I r

=

1

true

1 30 I

true

1 I 33 I

true

1 I 34- II 34

true

16

1 37 1 I 29 I r

=

1

1 31 I

true

1 | to I i 32

true

17

Tokens from places 1 , 1 , 26 30 with the

true

1

, and 1 33

get into place 34-

characteristics of chronic Kidney failure.

1 and 1 31 32

the tokens get into place 1 37

1 36

From places 1 , 29

with the characteristic of

acute kidney failure. The above constructed GN describes the a patient presents high level of BUN. K K

It

diagnostic process when

252

CHAPTER 4

§ 4. i. 5 Sena]

colic

The diagnosis of renal

colic is modelled in

the framework of

this GN. The tokens (patients) get into place 1 with the initial chai racteristic "symptoms of renal colic". Then they split up and get into the sections with initial places 1 and 1 and the appropriate charac2 3 teristics - investigation symptomatic

treatment

of

urine specimen

(the subside

and the

of renal colic),

result of

the

because of the

transition condition 1 2 r =1 1 1 1 1 Depending on the values of

1

true

3

true

these characteristics they

continue their

way in the GN. The transition :ondition

r 2

1 4

1 5

= 1 1 2 I

true

true

true .

1 I 31 1

true

true

true

The tokens in places 1 and 1 split 2 31

1

6

and

subsequently

places 1 , 1 and 1 , with the corresponding 4 5 6 the

their

characteristics:

They advance further depending on the values of the transition

conditions r , r , r and r : 3 4 5 6

r -- 1 1 3 41

1 7

1

W

W 2

1

8

1

1 9 W

3

lO W

4

where 1

into

results of macroscopic, test-strip and cytobacteriologic examina-

tions.

W

get

= "the urine is without any microscopic change";

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

253

254-

W

CHAPTER 4

= "there are heiraturia and/or blood coagulums" 2

(the token enters the GN from § 4. 1. 2); W

= "the urine sample is turbid" 3

(the token enters the GN from § 4. 1. 3); W

: "a calculus is emitted"; 4

1 1 11 r = 1 I true 4 10 10 I 1 1 1 12 13

r

= 1 I W 5 5 1 5

W

1 14

W 6

15 W

7

8

where z "there are no changes";

W 5 W

= "there is proteinuria" 6

(the token enters the GN from § 4. 1. 1); - "there is blood in urine"

W 7

(the token enters the GN from § 4. 1. 2); W

= "nitrites are present";

a

1 116 16 1 I true 7 I r = 1 I true 6 12 I 1 I true 5 I The transfer of tokens into place 1

means that a cytobacteri-

16 ological

examination is performed,

of the urine's sediment (1

which is a parallel investigation

) and microbiology (1 ) because 17 18

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

1 17 17 r 7 7

= 1 | true 16 16 I I

255

1 18 18 true

The transition conditions r , r and r have the forms 8 9 10 1

1 19

r = 1 | W 8 17 I 9

1

1 21

20 W

W lO

1 22 W

11

23 W

12

13

where W

= "the urine sediment is normal"; 9

W

= "there is heroaturia" 10

(the token enters the GN from § 4. 1. 2); W

= "there is leucocyturia" 11

(the token enters the GN from § 4. 1. 3); W

= "there is crystaluria"; 12

W

= "there is bacteriuria"; 13 1 24 24r = 1 r I t true rue 9 9 2 23 3I

r r

1 1 25 2 5

1 26

I W 24 I 14

W 15

= 1

10

;

where W

= "there is no bacteriuria"; 14

- -m .

w 15

14 The token's characteristic in place

infection is present" and in place 1 , 26

1 25

is

"appropriate

"no urinary tract therapy must be

256

CHAPTER 4

undertaken". The tokens in

places

1 and 3

1 split and get 22

on parallel

ways. The transition conditions r

, r 11

, r and r have the forms 12 13 14

1 1 1 27 28 r

1 29

30

= 1 | true 11 3 I

true

true

true

1 I true 19 I

true

true

true

r 12 12

1 31

1 32

= 1 I W 27 I 16 27 I 16

W 17 17

where W

= "the transfer 16

W 17

1 -> 1 is previously acconplished"; 3 27

= iW ; 16

r 13

1 33

1 34

= 1 I W 28 I 18

W 19

where W

= "there is no decrease in diuresis"; 18

W 19

= iW 18

(the token enters the GN from § 4-. 1.4);

r 14

1 35

1 36

= 1 I W 29 I 20

W 21

where W

= "there is no abnormal increase in body temperature"; 18

w

= nw . 21

20

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

257

The transition condition

r 15

1 37

1 38

1 39

1 I true 22 I

true

true

= 1 I true 11 I

true

true

1 | true 32 32 I I

true

true

1 I true 30 I

true

true

Therefore, when a token gets into place 1 , 1 , 1 or 1 it 11 22 30 32 splits into three. The transition conditions

r

, r 16 17

the forms:

r 16 16

1 40

1 41

= 1 I W 37 37 I I 22 22

W 23 23

where W

- "the renal function is normal"; 22

w

- -m

23 22 (the token enters the GN from § 4. 1.4);

r 17 17

1 42

1 43

= 1 I W 38 38 I I 24 24

W 25 25

where W

= "the X-ray of the urinary tract is normal"; 24

W = nW ; 25 24

r 18 18 where

1 45

1 46

= 1 I W 39 39 I I 26 26

W 27 27

r 20

have

258

W

CHAPTER 4

= "the metabolic functional tests are normal"; 26 - iV

W

27

;

26 1 44 r 19

- 1 ; I true 43 I 1 47

r 20 When

tokens

= 1 I true 46 I

get into places 1 , 1 , 1 ,1 ,1 ,1 ,1 , 8 9 13 14 20 21 34

1 and 1 , they leave this GN and get to the GN from § 4. 1.4. 36 41 In places 1 , 1 , 1 , 1 and 1 they end their movement in 25 33 35 42 45 the net with the characteristic "normal findings", ces 1 , 1 and 1 end with the characteristic 26 44 47

and tokens in pla­ "directing towards

appropriate treatment". x it

§ 4. 1. 6

*

Dysui-ia In normal

the ability

micturition cycle,

the continence

phase depends on

of the bladder adapt itself to urine volume

and maintain

pressure in urethra. During micturition, the bladder contraction asso­ ciated with sphincter relaxation empties the bladder. difficulty or pain associated with voiding.

Dysuria denotes

It may result from a wide

variety of pathological conditions and it is generally due to a cervico-urethro-prostatic obstacle. When this it is not the case one should investigate for a functional obstruction of the striated sphincter and the cervix, or a bladder hypocontractility induced by denervation, in­ hibition or muscular insufficiency.

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

The tokens (patients) get into place 1 with 1 racteristic "dysuria". Next they split up and get and

1 4

259

the initial cha-

into places

so that in each place they get the characteristics:

1 ,1 2 3 "results

from the case history" (1 ), "general examination" (1 ) or "complemen2 3 tary laboratory investigations" (1 ), because 4 1 2 2 r 11

= 1 I true 11 I I

1 3 3

1 4. 4

true

true

The transition conditions r , r and r have the forms 2 5 4 11 55

rr

= = 22

11 66

i1 iI w W 2 | | 11 2

w W 22

11 33

W W

II W W 11 11 II 11 II ffalse alse 17 II 17

11 II ffalse alse 55 55 II

2 true true true true

where W

= "there is a certain disease that may cause dysuria"; 1

w = -m ; 2

1 1 1 9 9 rr 55

= 11 II W W = 66 11 33

11 11 11 110 0

W w 44

W W 5

where W

= "consultation with a specialist is needed"; 3

W

= "consultation with a neurologist is needed"; 4

260

CHAPTER 4

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

W

261

= "urodynamic investigation is needed"', 5 1 12 r = 1 | true 4 5 I 1 I true 9 I From places 1 and 1 5 9

following characteristic:

the tokens get

into place 1 with the 12

"a specialist's opinion is required"

urologist, gynecologist, etc. ). From there the

(e.g.

token gets back to 1 , 6

passing through place 1 or to 1 according to the condition 17 16 1 16

1 17

r = 1 I W 8 12 I 6

W 7

where W

= "a definite cause of dysuria is found", 6

w = nw . 7

6 The transition condition 1 13 r 9 9

= 1 I W 11 11 I I 8 8

1 1 14 15 W

W 9 9

10 10

where W

= "the urodynamic tests are normal"; 8

W

= "the urodynamic tests are pathologic"; 9

W

= "the urodynamic tests are inconclusive". lO When tokens reach place 1 , 15

they leave

the net with the cha-

racteristic "further medical observation is needed".

262

CHAPTER 4

The transition condition 1 IS 18 r 10

= 1 I lO I

1 19

W

W 12

11

1 I false 13 I

true

where - "a neurological condition

W

that explains

the micturition disor-

11 ders is present";

- -m .

w 12

11 From place 1 (through place 1«

1 ), 35

the token

leaves the net

with the final characteristic "the micturition disorder is symptomatic of

a neurological condition".

Depending on the need

consultation (the transition r

of

psychiatric

has a form: 15

r 15 15

1 20 20

1 21 21

= 1 I W 19 I 13 19 I 13

W 14 14

where W

= "the patient must be consulted by a psychiatrist"; 13

W

= nW 14

13

). The tokens get into place 1 or 1 . From place 1 , the 20 21 20

token gets into place 1 or 1 (the transition r has a form 22 23 10

r lfl 18

1 22

1 23

= 1 | W 20 | 15

W 16

where W

= "the micturition disorder is of functional or anorganic origin"; 15

w

= nw ). 16

15

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

263

The transition condition 1

1 24

r

=

l | 14 I

16

W 17

1 25

W

1

1 27

1 28

29

W

W 20

W 21

22

26 W

18

19

where

W

= "vesical hypercontractility is present"; 17

W

= "a clearcut

pathologic

vesico-urethral dysfunction is manifes-

18 ted"; W

= "vesico-sphincteric dyssynergism is present"; 19

W

r "cervical obstruction is detected"; 20

W

= "there is vesical hypocontractility"; 21

W

= " other findings are present". 22 The transition conition 11 330 0 rr 114 4

= = 11 II W W 16 II 223 16 3

11 31 31

11 32 32

W W W W 25 224 4 25

where W

= "there is a definite diagnosis"; 23

W

= "there are uro-gynecological disorders"; 24

w 25

= nw . 24 Thus, from place 1 the token gets into places 1 , or 16 31

or 1 depending on 30

the characteristics it has acquired

(there is or is not uro-gynecological disorder, or another

1 , 32

in place 1 »2 disease is

present). When a token splits and engenders tokens in places

1 and 2

1, 3

264

CHAPTER 4

and when at a given moment these newly engendered tokens get into pla­ ces 1

, 1

31

and 1

21

(i. e. there is an accumulation of data for a gi24

ven patient from different channels), these three tokens fuse into one in place

1

with the characteristic

"micturition disorders of uro-

33 gynecological origin", because of the transition condition 1 33 1 I 21 I r = I 19 1 I 31 I 1 I 24 I

true true true

The transition condition 1 34

-=

r 21

1 I 32 I

true

1 I 23 | I 1 I 24 I

true

1 I 25 I

true

true

Therefore, when tokens engendered by an initial one simultaneo­ usly get into places 1 , 1 , 1 32 in place 1 34

with

23

and 1 25

,

they fuse into one

token

26

the characteristic "micturition disorders of

the

isolated neurogenic bladder". From 1 the token leaves the net with the characteristic "non29 dysuric disease". The transition condition

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

265

1 35 1 I 30 I

true

1 I 10 18 I|

true

1 I 22 I r = | 25 1 I 33 I

true

1

Therefore,

true

I 34 I

true

1 I 27 I

true

1 I 28 I

true

1 I 54 I

true

from each of the places

and 1 , the token gets into 1 54 35

1 ,1 ,1 ,1 ,1 ,1 ,1 30 18 22 33 34 27 28

which symbolizes

the performance of

an appropriate treatment. The transition condition 1 7 r 3

- 1 l 4 I

true

1 8 true

The token from 1 splits into two new tokens which get into 1 (inves4 7 tigation for hyperazotemia) and 1

(urinalysis).

8 The transition condition

r 6

= 1 I 7 I

1 136 36 W 26

1 1 37 37 W 27

where W

= "elevated blood urea nitrogen is present"; 26

266

w

CHAPTER 4

= nw 27

26 From 1 the token gets into place 1 or 1 according 7 36 37

dition r 7

to con-

(hyperazotemia is present or is not). In the first case the

token leaves the net and gets into the GN from § 4. 1. 4. From 1 , the token splits into three which 0 to places: 1 ("two glasses" test), 38 ons, including for b. Koch) and 1 40

1 39

(bacteriologic investigati-

(micturition flow), because

1 38

1 39

1 40

r = - 1 I true 7 7 I

true

true

The transition conditions r

, r 11

r 11 11

get correspondingly

and r 12

= 1 | 38 38 I I

have the form 13

1 41

1 42

W 28 28

W 29 29

where W

= "turbid urine is found"; 28

= -m

w 29

28

1 1 43 r 12

= 1 I 39 I

W 30

1 1 44 W 31

where W

= "a positive bacteriologic finding is present"; 30

w 31

= nw 30

APPLICATIONS OF GENERALIZED NETS IN HEDICINE

1 45 45 r 13 13

s 1 | 40 I 40 I

W 32 32

267

1 46 46 W 33 33

where W

= "there is an increased urine flow"; 32 = iV

W 33

32 The increase of urine flow means polyuria, so the token in pla­

ce 1 leaves the net with this characteristic. 45 From places 1 , 1 , 1 and 1 the tokens get into place 1 37 42 44 46 48 (combined excretory and mictional urography) because 1 48 1 | true 37 I r = 1 I true 20 42 I 1 I true 44 I 1 I true 46 I The transition conditions r , r , r and r have the form 23 24 25 22 1 49 49 r = 1 I W 22 48 22 48 I I 34 34

1 50 50

1 51 51

W

W 36 36

35 35

where W

= "the X-ray picture reveals distinct pathological finding", 34 - IV

W

35 W

,

34 = "the X-ray report is in conclusive".

36

268

CHAFFER 4

r

= 1 I 23 51 I

1 52

1

W

W 37

53 38

where W

= "suprapubic cysto-urethrography is performed"; 3?

W

= "retrograde urethrography is performed" ; 38

r 24

1 54

1

1 I 49 I

W

W 40

= - 1 I 50 I

W

1 I 52 I

W

1 I 53 I

W

39

W 39

40

39

W 40

39

W 40

where W

= "there is a definitive diagnosis"; 39 - iW

W 40

. 39

Finally, 1 47 r

= 17

1 I 41 I

true

1 I 43 I

true

K

It

It *

55

APPLICATIONS OF GENERALIZED NETS IN IVEDICINE

§ 4. 1. 7 Arterial

269

hypertension

The purpose of the diagnostic examination of a patient

is with

hypertension to determine both the severity of the disease and whether the hypertension is primary (essential) or secondary to another disor­ der. The results of this procedure

will determine the need for treat­

ment as well as the type of treatment. The systemic arterial hypertension is usually astolic pressure

defined as a di-

(DBP) of 90 nnHg or higher recorded on

minations of a subject with normal salt intake

several exa­

( 1 0 + 3 g/day) regard­

less of age and sex. A token in place 1 means a patient 1

with arterial hypertension

(as initial characteristic). The transition conditions of the GN have the form 1 2

r 1

- 1

I W 1 1 1

1

1

1

3

4

W 2

W 3

1 5

W 4

6

W 5

where

W

= "recording the DBP under appropriate conditions"; 1

W

= "fundoscopic grading of retinal changes

(according Keith-Wagner-

2 Barker classification)"; W

= "each of left ventricular hypertrophy"; 3

W = "investigation for renal damage"; 4 r

- "evidence

of target organ damage or

presence

5 risk factor other than hypertension";

r 2 where

1 7

1 8

1

= 1 I W 2 1 6

W 7

W 8

9

of cardiovascular

2T0 270

CHAPTER 4

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

W

= "the DBP is within the limits 90-1O4 imHg"; 6

w

= "the DBP is within the limits 105-114 mnHg"; 7 = "the DBP is i 115 imHg";

W ft

1 10 10 r 3

= 1 I W 3 1 9

1 11 11

1

W

W w 11

12 12

10

where W

= "no retinal changes or such of KWB group 1"; 9

W

= "retinal changes KWB group 2"; lO

W

= "retinal changes KWB groups 3 and 4"; 11 1 13

1 1 14 15

r = 1 I W W 4| 12 13 4

W 14

where

- -m

W 12 W

& nW ;

13

14

= "paterns of left ventricular 'strain'"; 13

W

= "left ventricular hypertrophy is present"; 14 1 16 16 r 5

1 17 17

= 1 I W W 15 16 5 I

where W

= "no renal damage"; 15

W 16

= nW ; 15 1 1 1ft 18 1 19 9 r = 1 I W r 6 6 7 6 6 II 117

W 1ft 1ft

271

272

CHAPTER 4

where W

= "no evidence of other target organ damage

or other cardiovascu-

17 lar risk factors"; W

= nW 18

; 17

- 1

r 7

1 20

1

I W 10 19 10 I

W

21

20

where W

= "there is a token, related to the present token, at least in one 20 of the following places: 1 , 1 , 1 , 1 ,..., 1 , 1 "j 8 9 13 14 17 19

= -m ■,

w 19

20

r

= 1 8

1 22

1

I W 13 I 21

W

23 22

where W

= "there is a token, related to the present token, at least in one 22 of the following places: 1 , 1 , 1 8 9 = iV

W 21

,1 11

,1 ,1 "; 12 17 19

; 22

r

= 1 9

1 22

1

I W 16 23 16 I 23

W

25 24

where W

= "there is a token, related to the present token, at least in one 24 of the following places: 1 , 1 , 1 , 1 , 1 , 1 ■'; 8 9 13 14 15 19 = iV

W 23

; 24 1 26 r

= 1 I W 17 I 10 25

1 27 W 26

APPLICATIONS OF GENERALIZED NETS IN MDICTNE

273

where W

26

= "there is a token, related to the present token, at least in one

of the following places: 1 , i , i , i "; 9 12 16 19 W = iH ; 25 26 1 1 28 29 r

11

= 1 | W 27 19 1

W 28

where W

28

= "there is a token, related to the present token, at least in one

of the following places: 1 , l , 1 , 1 "; 9 12 15 17 W = nW . 27 28 The remaining transition conditions' predicates

(of the 12-th,

13-th and 14-th transitions) are only "true". K K

§ 4. i. 8 Arterial

hypertension

«

of renal

The tokens get into place 1 1

origin

with initial characteristics for-

med by three groups of data: - group X: "signs and symptoms

directing to renal origin of the arte-

rial hypertension" - group Y: "signs and symptoms directing to renovascular origin of the arterial hypertension" - group Z: primary

"there are no signs, symptoms or conditions, excluding the participation of kidneys in the

origin of the hypertensive

state". Hie transition condition

r

1 2

= 1 1 true 1 11

1

3

true

274-

(

CHAPTER 4

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

275

Thus the token splits into two, which get into place 1 (urine is sent 2 for analysis)

and place 1 3

(blood sanple is taken

for examination).

The token in place 1 splits into three new tokens that simultaneously 2 get into places 1 (biochemical examination of urine), 1 (urinary se4 5 diment examination) and sition condition r

1 6

(bacteriologic examination) because tran-

has the form 2 1 4

1 5

r = 1 I true 2 2 I i 2 2

1 6

true

true

The transition conditions r and r have similar form 4 5 1 10 r = 1 I true 4 4 I 1 1

12 12

r

= 1 I true 5 5I

when the token gets into places 1

true ;

1 1 1

1

13 13

true

14 14 true

, . . . , 1

10 taneously for proteinuria,

1 11

,

if means testing simul-

14

glucosuria,

hematuria,

leucocyturia

and

bacteriuria. The transition conditions r , r , r 6 lO 11

r = 1 I 6 6 1

1 15

1 16

W

W 1

where W

= "there is positive urine culture", 1

2

r

have the form 14

276

W

CHAPTER 4

= nW ; i

2

1 ) 24 24 r 10

= 1 I W 3 10 I

1 25 25 W 4

where W

= "there is protenuria"; 3

W

-- nW ; 3

4

1 26 r

= 1 I W 11 10 I 5 11 10 I 5

1 27 W 6 6

where - "diabetes mellitus is suspected";

W 5 W 6

= lW ; 5 1 28 28 r 12 12

= 1 I W 12 I 7 12 I 7

1 29 29 W 8 S

where W

= "there is hematuria"; 7

w = nw ; a

7 1 l 30 30 r 13

= 1 | W 13 I 9

l 1 31 31 W 10

where W

= "there are bacteria in urine sediment"; 9

W 10

= nW ; 9

APPLICATIONS OF GENERALIZED NETS IN MBICTNE

1 32 r

= 1 | W 14 1414 I 11

277

1 33 W 12

where W

= "there is bacteriuria"; 11

W 12

= iW ; 11 The transition condition

r

= 17

1 36

1

1 I 25 I 25 I

W 13 13

W 14 14

1 I 27 I | 1 | 29 I

W 13

W 14

W

1 I 31 31 I I

W

1 I 33 I 33 I

W

1 16 16 | |

W

37

W 13

14

13 13

W 14 14

13 13

W 14 14

13 13

W 14 14

where W

= "glomerular nephropathy is suspected"; 13

w

= nw . 14

13 The token in place 1 splits into three tokens which get into 3

places 1 (serum creatinine by urea determination), 1 (serum potassi7 8 urn), and 1 (blood sugar), because the transition condition r has the 9 3 form: 1 7 r 3

= 1 I true 3I

1 8 true

1 9 true

278

CHAPTER 4

The transition conditions r , r , r , r and r have the form 7 ft 9 10 17

r 7

1 17

1

= 1 I W 7 I 15

W

18 16

where W

= "there

is increase in serum creatinine level";

15

w

= nw 16

15 1 19 19 r = 1 I W ft 17 ft ft ft I I 17

1 20 20

1 21 21

W

W 19 19

18 18

where W

= "hyperpotassemia is present"; 17

W

= "nomaopotassemia is present"; 18 - "hypopotassemia is present";

W 19

r 9 9

1 22

1

= 1 | W 9 20 9 I I 20

W

23 21 21

where W

= "there is hyperglycemia"; 20

W 21

= iW ; 20 1 34r

= 1 | W 18 I 22 10

1 35 W 23

where W

= "there is a low glomerular filtration rate"; 22

w 23

= iw 22

;

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

279

1 39

Al1 tokens place

1 39

1 I 37 I

true

r - 1 | 17 35 I

true

1 I 21 I

true

1 I 23 I

true

pertaining to a specific patient 'when getting

into

fuse into one token with characteristics incorporating all

the previously

acquired characteristics by the tokens

the investigations carried out up to now. Place 1

resulting from

symbolizes "a test

39 for iodine sensitivity is performed". The transition conditions r 19

r 19

and r have the form 20 1 1 1 1 40 41

= 1 I W 2439 I 24

W 25

where W

= "the test for sensitivity is negative"; 24-

W 25

= nW ; 24

r 20 20

1 42

1 43

= 1 I W 40 40 I I 26 26

W 27 27

1 | false 41 I

true

where W

= "intravenous urography is performed"; 26

W

= "other 27

instrumental

investigations

of kidneys

are

necessary

280

CHAPTER 4

(isotopic, sonography, etc. )". The place

1

corresponds

to the following actions: isotopic

43 investigations (renography, scintigraphy,

gamma-camera >,

sonography,

computer tomography, and the place 1 to excretory urography. 42 The transition conditions r , r and r have the form 21 22 23 1 1 44 45 44 45 r = 1 I W 21 42 21 42 I I 28 28

W 29 29

where W 28

= nW ; 29 - "there are urographic data for renovascular hypertension";

W 29

1 46 46

1 47 47

r = 1 | W 22 43 22 43 I I 30 30

W 31 31

where W 30 W

= nW ; 31 = "there are signs of renovascular hypertension";

32 1 48 48

1 49 49

r = 1 I W 44 I 32 23

W 33

1 I W 46 I 32 46 I 32

W 33 33

where W 32 W

= nW ; 33 = "the initial

characteristics

of the token

has positive

siqns

33 from group Y". When tokens get into place

1 24

there is

a call

to

GN

from

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

§ 4. 1. 3, in place 1 28

281

- to GNs from § 4. 1. 2 and § 4. 1. 5, in places 1 15

and 1 to GNs from § 4. 1. 3 and § 4. 1. 5; in places 1 , 1 30 17 19 GN from § 4. 1.4; in place 1 - to GN from § 4. 1. 3; 32

and 1 34

in places 1 47

to and

1 - to GN for § 4. 1. 9. 49 The tokens in places 1 , 1 and 20 22

1 26

leave the GN because of

the indications for other diseases, and in places 1 and 1 they re38 48 ceive the characteristic: appropriate treatent and medical observation of the patient.

* X

§ 4. i. 9 Renovascular The tokens

*

hypertension (patients) get into place 1 from a GN from § 4. 1. 8 1

with the initial characteristics

"signs and symptoms directing to re-

novascular origin of the arterial hypertension", The transition condition r

has the form: 1

1 2 r

- 1 1 true 1 11

A token from place places

1

1 1

1

3

true

1 4 true

5 true

splits into four tokens

1 ,..., 1 . These places signify: 2 5

that get

into

1 - check-up for endocrine 2

ne disorder (primary hyperaldosteronism, pheochromocytama, hypercorticism etc. ), 1 - performing an iodine sensitivity test, 3 gation

of

plasma renin activity, 1

tests. The transition condition

1

4

- investi-

- performance of pharmacological

282

CHAPTER 4

1 6 r 2

1 7

= 1 I W 2 | 1

W 2

where W

= "an endocrine

disease as

a probable cause

of the

hypertensive

1 state is found", W 2

-- -W . 1 Place 1 is the final one for the net. 6 The transition conditions r , r , r , r , r 3 4 5 8 9

and

r 6

have the

forms

1 3

1 8«

1 9

II 11 rr = II 3 11 II 10 II 10

W W 33

W W 4

W W 3

W W 4

1 II 17 I 17 I

W W 3

W W 4

1 II 19 I 19 I

W W 3

W W 4

where W

= "pharmacological test gives negative result and it is possible to 3 perform a vasography";

w

= w 4

; 3

- "high

W

level of plasma renin activity is found";

5 W 6

= nW

; 5

1

1 12

r

= 1 5

where

I W 5 | 7

13 W 8

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

W

283

= "angiotensin test is performed"; 7

W

= "test with saralasin is performed";

a

1 1 17 17 r d 8

= 1 | W w 12 12 I I 9 9

1 1 18 18 W w 10 10

where - "the angiotensin test is positive";

W 9 W

10

= nW ; 9 1 19 19 r - 1 I W 13 I 11 9

1 20 20 W 12

where W

= "the saralasin test is positive"; 11

W 12

= nw ; 11 1 14 r = 1 I 6 7I

true

1 | 8I

true

The tokens from places 1 and 1 reach directly place 1 (per7 8 14 forroance of

vasography - abdominal aortography and/or selective renal

arteriography). The transition conditions r , r , r and r have the form 71 10 11 12 1U 11 1-

r 10 where

1 21 21

1 22 22

= 1 I W w 14 I 15

w 16

284

W

CHAPTER 4

= "there are signs of renal arterial stenosis"; 15

W 16

= nW ; 15

r 11 11

1 23

1 24

= 1 I W 21 I 17 21 I 17

W 18 18

1 | true 15 I 15 I

false

where W

= "there is

evidence that

the stenosis is

functionally signifi-

17 cant", = iW ;

W

18

17 1 25 r

= 1 | true 12 23 I

Places 1 , 1 25 26 tests to

26

1

1 27

1 28

29

true

true

true

true

1 indicate the performance of appropriate 29

determine whether

1 (Howard's test), 25

1

the stenosis has functional significance:

1 (Rapoport or Stanley test), 1 (separate he26 27

modynamic investigation of Kidneys), 1 (determination of 28 nin activity in renal

plasma re

veins separately), 1 (other investigations of 29 separate Kidney functions).

r 13

1 30

1 31

= 1 | W I 25 | 19

W 20

where W

= "the test of Howard is positive", 19

w 20

= -m ; 19

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

r 14 14

1 32

1 33

= 1 | W 26 26 I I 21 21

W 22 22

285

where W

= "the test of Rapoport or Stamey is positive"; 21

W 22

= nW ; 21 1 34 r

= 1 I W 15 27 I 23

1 35 W 24

where W

= "unilateral decrease of ERBF is found during

separate hemodyna-

23 mic isotopic investigation of kidneys"; w

- -m ■, 24 23 1 36 r 16 16

= - 1 I W 28 2ft I I 25 25

1 37 W 26 26

where w

= "augmented plasma renin activity is determined unilaterally"; 25

W 26

= nW ; 25

r 17

1 38

1 39

= 1 I W 29 I 27

W 28

where W

= "other functional 27 on"; 28

27

tests for

determining unilateral renal lesi-

286

CHAPTER 4

1 40

1 41

1 I 30 I 30 I

W 29 29

W 30 30

1 I 32 I r = I 18 1 I 34 I

W 29

W 30

W

W 29

30

1 I 36 36 I I

W 29 29

W 30 30

1 I 38 I 38 I

W 29 29

W 30 30

where - "a biopsy of the contralateral kidney is necessary";

W 29 W 30

= nW . 29 Place 1 signifies the performance of renal biopsy. 40

r 19 19

1 42

1 43

= 1 I W 40 40 I ! 31 31

W 32 32

where W

= "evidence for lesion of the contralateral kidney is present"; 31

W 32

= nW . 31 1 44 r 20

= 1 | true 43 I 1 I true 41 I

Place 1 signifies aa check-up for contraindications to surgi44 cal intervention. The transition condition

r 21

1 45

1 46

= 1 I W 44 I 33

W 34

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

287

288

CHAPTER 4

where W

= "there are contraindications"; 33

W 34-

= nW 33

(a surgical investigation (1 ) is possible). 45 From

places

1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 1 , 11 16 18 20 22 24 31 33 35 37

1 , 1 and 1 the tokens get into place 1 . Here, they acquire the 39 42 46 47 characteristic

"appropriate treatment and medical observation

of the

patient".

REFERENCES'. [1] F. Balag, G. Petranyi and F. Renyi-Vamos,

Nephrologia,

Medicina

Acute renal failure,

Saunders

Konyvkiado, Budapest, 1980 (in Hung. ). [2] B. Brenner and J. Lazarus (Eds. ), Co. , Philadelphia, 1983. [3] B. Brenner and F. Rector (Eds. ), The kidney, Vol. 1 and 2,

Saun­

ders Co. , Philadelphia, 1986. [4] N. Bricker and M. Kirschenbaum (Eds. ), The kidney - diagnosis and menagement, Wiley Medical Publ. , New York, 1984. [5] J. A. Halsted and C. H

Halsted (Eds. ) The laboratory in clinical

medicine - interpretation and application,

Saunders Co. , Phila­

delphia, 1981. [6] J. Hamburger, J. Crosnier and J. -P. Grunfeld (Eds. >, Nephrologie, Vol. 1 and 2, Flanmarion, Paris, 1979 (in French). [7] J. Jones, J. Briggs and T

Hargreare, Diagnosis and menagement of

renal and urinary diseases, Blackwell, Oxford, 1982. [8] S. Klahr and S. Massry, Contemporary Nephrology, Plenum Publ. Co. New York, 1981. [9] H

Kremling, W. Lutzeyer and R. Heintz,

Gynakologische

Urologie

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

289

und Nephrologie, Urban Schwarzenberg, Munchen, 1982 (in German). [10] M

Legran and J. -M Sue,

Abrege de nephrologie,

Masson,

Paris,

1981 (in French). [11] G. Majdrakov and . Popov (Eds. ) Gidney diseases, Sofia,

Medicina

i Fizkultura, 1980 (in Bulg. ). [12] F. Marsh (Ed. ) Postgraduate nephrology,

W. Heinemann Medical Bo­

oks Ltd. , London, 1985. [13] S. Massry and R. GlassocK, Textbook of nephrology,

Vol. 1 and 2,

Williams and Wilkins, Baltimore, 1983. [14] E. Patev, Acute renal failure, Sofia, Medicina i Fizkultura, 1981 (in Bulg. ). [15] D. Vasilev, Vasorenal hypertension, Sofia, Medicina i Fizkultura, 1989 (in Bulg. ). [16] D. Vasilev and G. Stefanov

(Eds. ),

Clinical nephrology,

Sofia,

Medicina i Fizkultura, 1990 (in Bulg. ). [17] J. Sorsich and K. Atanassov.

Application of generalized

medicine (Acute Attack of Gouty Arthritis),

nets in

Proc. of Third Symp.

Int. Ing. Biomed. , Madrid, Oct. 1987, 643-64-5. [18] R. Bellman, Mathematical methods in medicine.

World Sci. Publ. ,

1983. [19] S. Kiforenko, Methods of mathematical biology, Kiev,

Vischa sko-

la, 1984 (in Russian). [20] J. Piemann, Modeling hospital information systems with Petri nets Methods Inf. Medicine, Vol. 27, No. 1, 1988, 17-22. [21] Mathematical methods

in medicine

and biology,

Proc. of

UNC of

Academy of Sciences of USSR, Sverdlovsk, 1986 (in Russian). [22] J. Sorsich,

An example of application of generalized nets in me­

dicine. Proc of II Int Symp. "Automation

and Scientific

Instru­

mentation", Varna, May 1983, 387-389. [23] J. Sorsich and K. Atanassov, medicine

Application of generalized nets

in

(Diagnostics of arterial hypertension of renal origin).

290

CHAPTER 4

Proc. of III Int. School

"Automation and Scientific Instrumenta­

tion", Varna, 1984, 233-236. [24] J, Sorsich and K. Atanassov,

Application of generalized

nets in

medicine (Renal colic), Lect. Notes in Medical Inform 24, 1984, 352-355. [25] J. Sorsich and K. Atanassov, medicine

(Renovascular

Application of generalized nets

in

hypertension). Proc. of Third Int. Symp.

"Automation and Sci. Instrumentation", Varna, 1985, 167-169[26] J. Sorsich and K. Atanassov,

Application of generalized nets

in

medicine (Permanent Proteinuria), Proc. of National Sci. Session "Automation of Biotechnology Processes",

Sofia, Oct. 1985,

138-

141 (in Bulgarian). [27] J. Sorsich and K. Atanassov,

Application of generalized nets

medicine (Haematuria), Proc. of III International Symp.

in

"Automa­

tion and Scientific Instrumentation", Varna, Oct. 1985, 163-166. [28] J. Sorsich and K. Atanassov,

Application of generalized nets

in

medicine (Dysuria), Proc. of III Symp. Int. Ing. Biomed. , Madrid, Oct. 1987, 639-642. [29] J. Sorsich, Application of the generalized nets in medicine (Ele­ vated blood used nitrogen),

First Sci. Session of

the Math. Fo­

und. of AI Seminar, Sofia, Oct. 10, 1989, Preprint IM-MFAIS-7-89, Sofia, 1989, 57-59. [30] J. Sorsich,

Application of the generalized nets in medicine (Ar­

terial hypertension), Ninetieth Session of

the Nat. Seminar

Informatics of the Union of Bulgarian Mathematicians

and

of

Fourth

Sci. Session of the Math. Found. AI Seminar, Sofia, Nov. 5, 1990, Preprint IM-MFAIS-5-90, Sofia, 1990, 37-39.

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

291

§4.2: MODELLING OF DIAGNOSTIC AND THERAPEUTIC PROCESSES IN MEDICINE BY GENERALIZED NETS Borjana Jordanova

and

In this article we describe

Joseph Sorsich

a general outline of

the approach

to a patient who needs medical advice. This net is more general than the ones lacks the details of the others, that most patients pass through

described in § 4. 1.

It

but it gives some idea of the stages in a medical department.

So the nets

described in § 4.1 are its concrete applications. In the

net presented it is possible to

see more

clearly the

ways of introducing waiting and performing times of medical examinati­ ons

and carrying out the

time may

appropriate investigations,

duration of the whole cycle of treatment.

as well as the

These time-parameters

be obtained by a characteristic function of

the net as presented

below. The GN models

of

the diagnostic process

and

the appropriate

treatment of other (non-nephrological) diseases may also be settled in this scheme. The token

(patient)

enters place

1

with the characteristic

1 "need to consult a physician"

(with some complaints or for prophylac­

tic needs). The GN below described is a reduced GN from the class A ,A ,A ,A ,n ,n ,c,6 ,6 , n ,G ,b 3 4 6 7 A L 1 2 K K

I

(everywhere the transition type is "v", ties are

"
the places' and arcs' capaci­

the places' and transitions' priorities are "0",

every token a: b(a) =
is 1

for

292

CHAPTER 4

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

Z

=
,

1

1

),

(1 , 2

14

1 ), 3

293

r >, 1

where 1

1 2

r

=

1

1

I i

1

l

I 14 I 14 I

W

3 W

l

2 W 1 1

W 2 2

where

W

= "the result of the medical examination is that the patient is hei althy";

w = nw . 2

1 The ransition Z

2 2

is

Z == <[1 <[1 ,, 11 ,, 11 ,,11 ,,11 )), , {1 {1 ,, 11 ,, 11 1, 1, rr > >,, 2 2 33 66 lO lO 12 12 220 0 44 55 66 2 where 1 1 4 4

rr

= = 22

11 55

11 6

11 II W W 33 | | 3 3

W W 4 4

W W

11 II W W 66 | | 3

W W 4 4

W W 55

11 II W W 110 0 II 33

W W 44

W W

11 II W W 112 2 I I 33

W W 4 4

W W 55

11 II W W 220 0 II 33

W W 44

W W 55

55

5 5

where W

= nW 3

W

4

& nW , 5

= "there is need for appropriate investigations"; 4

W

= "there is need to consult a specialist". 5

294

CHAPTER 4

The transition Z

is 3 3

Z = <(1 >, <(1 i, i, (1 (1 ,, 11 1, 1, rr >, 33 55 77 88 3 3 where 1 77 rr

= 3

11 55

1 8 8

Il W W 11 66

W W 77

where W

= "laboratory investigations are necessary"; 6

W

= "instrumental investigations

(e. g. X-ray, sonography,

7 are necessary". The transition Z is 4 Z = <[1 <[1 ,, 11 1, 1, II II ,, 11 ), ), rr >, >, 44 77 88 99 10 4 10 4 where 1 99 rr = 4

1 10 10

11 77

II W W 11 88

W W 9

1 88

II W W 11 88

W W 9 9

where W

= "abnormal findings are present"; 8

w = nw . 9

a The transition Z

is 5

Z : <(1 , 1 I, 5 4 9 where

[1 11

,1 12

,1 13

,1 14

), r >, 5

ECG etc. )

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

1 11 r

= 5 5

1

1

1 13

12

1 4 4

I W I 10 I 10

W

1 9

I W I 10

W

295

W 11 11

14 W

12 12 W

11

13 13 W

12

13

where W

= "patient needs appropriate treatment"; 10

W

= "patient needs some complementary investigations"; 11 - "patient is estimated to be healthy";

W 12

- nw

w 13

& -m

10

& iv

11

. 12

The transition Z

is 6

Z = <(1 }, 11 , 1 , 1 ), r >, 6 11 15 16 17 6 6 11 15 16 17 6 where 1 15 r = 6 6

1 I W 11 14 11 I I 14

1 16

1

W

W 15 15

17 16 16

where W

= "ambulatory treatment is suitable"; 14

W

= "treatment at home is reccnmeanded"; 15

W

= "hospitalization is necessary". 16 The transition Z

is 7

-.
Z 7 where

,1 15

,1 i, [1 , 1 , 1 !, r >, 16 17 18 19 20 7

296

CHAPTER 4-

1 18 1 I W 15 I 17 r = I 7 1 I W 16 | 17 1 I W 17 17 17 I I 17

1 19

1 20

W 18

W 19

18

W 19

18 18

W 19 19

W W

where W

= "patient is restored to health"; 17

W

= "patient is dead"; 18

. iw

w 19

17

& -m . 18

The tokens receives in the different places racteristics by the characteristic function $ = §

the following cha-

20 U , defined by: i=2

-> "healthy person"; 2

$

-> "duration of medical examination"; 3

$ -> "K"; 4 §

-> "Kinds of appropriate laboratory examinations, the duration to 5 carry out these examinations";

$

-> "Kind(s) of appropriate consultation)s), duration of these con6 sultation(s)";

$

-> "result(s) of the laboratory investigation)s)"; 7

$ -> "result(s) of the instrumental investigation)s)"; 8 §

-> "presence of abnormal findings"; 9

$

-> "*"; 10

$

-> "patient needs appropriate treatment"; 11

APPLICATIONS OF GENERALIZED NETS IN MEDICINE

$

-> "patient

needs

complementary investigation(s)

297

or consultati-

12 tion(s)"; $

-> "patient is estimated healthy after the consultation(s)"; 13

$ -> "further surveillance or control is needed"; 14 $

-> "ambulatory treatment"; 15

$

-> "treatment at home"; 16

$

-> "treatment in hospital"; 17

$

-> "the patient is cured with

end of treatment,

duration

of the

18 treatment"; $

-> "exitus letalis"; 19

20 The characteristic function determines some

time-parameters of

the model. For their calculation statistical data,

as well as results

from concrete observations, can be used.

Chapter

5:

GENERALIZED NETS AND COMPUTER S C I E N C E

Some applications of the generalized nets as a means for model­ ling of computers are introduced in § 5, 1, § 5. 2 models are related to

the various

and § 5. 5.

The

processes in the computer and they

are written by the postgratuate students of

the Institute for Micro­

systems, Sofia. § 5.6 the

deserves special attention where a short

Program Package for Generalized Nets

scription corresponds

(PPGH)

to the current state of

description of

is given.

the work on

contains the basic directions of its further development.

298

This de­ PPGH

and

GENERALIZED SETS AND COMPUTER SCIENCE

299

§5.1: IMPLEMENTATION OF GENERALIZED NETS IN DISTRIBUTED OPERATING SYSTEH DEVELOPMENT NIKo Fetalivanov

§ 5. i. 1

Intro&ictian

A general view upon

the latest developments

in operating sys­

tems snows that distributed processing is a subject of tion.

This interest may be explained,

growing atten­

on the one band by the increa­

sing demands for more computing power and, on the other by cements to hardware technology.

The advantage of distributed

ting systems has now become a fact, in computer science,

its enhan­

after a long

particularly in the field

opera­

period of evolution

of operating systems.

In the last several years that local area networks (LANs) and personal workstations have become so popular and widely problems associated with these arise

and

used and that serious

required solution.

Present

operating systems provide time-sharing, multiprocessor, graphics, net­ working, etc. shared

However,

computing and

the answer

to

demands for efficient

storage resources is

in distributed

use of

operating

systems. The design of such distributed systems has proved to be and difficult process, which sometimes takes years is caused

by

ning on them

a long

of hard work. This

the presence of multiple nodes with separate tasks run­ that share system resources and require synchronization,

control and load balancing.

A major consequence of the latter

is the

numerous set of states in which the system may appear. Its performance is therefore determined by the ability of the software to act tely in each

adequa­

particular state, which itself is defined by the respec­

tive algorithms. A general way to facilitate software

design is

to present the

algorithms, the corresponding system states and the transitions betwe­ en

them in

a graph scheme.

The implementation

of

Generalized Nets

300

CHAPTER 5

(GNs)

is particularly suitable for this purpose,

as they

provide

a

universal method for simulation of system behaviour. The purpose of implementation in

this article is

one specific area

to outline some aspects of

of distributed

GN

operating system

development - the design of distributed file system (DFS).

§ 5. 1, 2 Generalized The DFS is

Net model for a Distributed an element of

File

every distributed

System operating system,

which carries out functions of vital importance. It has to provide the availability and consistency of all files, to access,

giving distinct

control concurrent file

response times to client requests.

To meet

these requirements, a DFS manages all shared storage resources, placed at the nodes, maintaining all data integrity and security. One generalized presentation of shows several nodes in rage resources

a DFS is given in Fig. 5. 1, It

a distributed system,

attached to them,

seme of which have sto­

while the others

do not have such.

All system services and hardware components that accomplish between

the nodes are presented as

the links

a cowminication medium,

carrying

out data transfer. This picture will make it easy to follow the client requests and server responses when the discussion

is

focused at the

GN model of the DFS. A common approach

to maintaining high availability

response time is the use of 'replicated' files. have

copies at multiple nodes across

their modification quite efficient.

does not

take

and

short

These are files that

the distributed

system

When

place frequently, this method is

However, attempts to modify a replicated file invoke

special synchronization mechanism, traffic

and

which

affects system performance.

causes

heavy communication

Ordinary files do

multiple copies and are processed in a simpler way,

not have

but their average

access time is usually higher. Normally

a

DFS allows

several operations

to be

carried out

GENERALIZED NETS AND COMPUTER SCIENCE

301

Fig. 5. 1 upon files, such as read, write, create, etc.

There is

a special way

in which all file modifications take place: they are either successfully, or the system is left unchanged. sociated with

this type

of action,

completed

A specific term is as­

reflecting

its nature - 'atomic

transaction'. Taking into consideration the above features of been possible to form

a

Fig. 5. 2.

DFS, it has

GN model of the distributed system currently

designed at the Institute of Microsystems, Sofia. in

a

The places in the

net represent

This model is shown the states,

in which

client requests appear, while they are being serviced by the DFS.

The

302

CHAPTER 5

toKen stands for

a

particular client request at

transitions of these tokens between the places

a

are

given time.

The

determined by the

respective predicates. An important advantage of GNs

is

their ability to assign cer­

tain characteristics to the tokens. They may be further used to analy­ se the events that have taken place in the DFS, their causes

and re­

sults. Some definitions have to be made prior to their use in predica­ tes and token characteristics. Let every file 'f

in the system have a

function 'r(f)' defined for it, such that r(f) = 1, if 'f is an ordinary file, and r(f) > 1, if 'f

is a replicated one,

On this basis, the sets 'Fo' and 'Fr' of ordinary and replicated files in the DFS are defined as follows: Fo = ( f / r(f) -- 1 !, Fr = ( f / r(f) > 1 ). Every token 'a' tic

entering the net is assigned

a x - "", 0

the initial characteris-

where 'clientnode'

which the respective request is serviced,

is the node

in

'f is the file identifier,

and 'o' is the operation to be applied on ' f . In places

1 and 1 the tokens are assigned new characteristic 3 5

expressed by the respective characteristic functions $ 3 $

and $ : 5

-> "<server_node>", 3 where 'servernode'

is a node in the system where the request

is to be serviced. $ -> "", 6 where 'trans_id' is a transaction identifier system;

'TIME' is the current time,

assigned by the

'6' is the period during

which the transaction is active, 'result' signifies whether the

GENERALIZED NETS AND COMPUTER SCIENCE

operation has or has not been successful: result € ( true, false ).

Fig. 5. 2. The t r a n s i t i o n c o n d i t i o n s are defined as f o l l o w s : 1 2 r

w*iere W ='f 1

E Fr";

W = "f € Fo"; 2

1

=

1

I W 1 1 1 I 1 I W 2 I 3

1 3 W 2 true

303

30*

W

CHAPTER 5

= oc a = "c(l , X ) < r(pr2 x )" 3 3 0 0

where

c(l, X)

being a function giving the number of tokens

in place

'1' with characteristic 'X';

rr -2

1 i 4 4 W W 44

1i

5 5 W W 5

1 II 33 || II 1 II ffalse alse 4 II

true true

where

4

a a = "x - pr x "; cu 10

5

= IW ; 4

W W

rr 3

::

1 6 6

11 77

1

1 II W W 55 || 66 II 11 II W W 6 1 9 6 1 9

W W 77

W W 8

W W 10 10

W W 11 11

88

where W

= "f e Fr"; 6

W 7

a a = "f e Fo" St "pr x i Tr" & "pr x = true" 3 cu 4 cu where 'Tr' being the maximum time allowed by

the system

which a transaction may be active, a a W > = W = = "f "f ee Fo" Fo" & & "pr "pr xx > Tr" Tr" vv "pr "pr xx = ff aa ll ss ee " " ;; 8 3 cu 4 cu 8 3 cu 4 cu = a a w = "c(l , x ) < r ( p r x )"; 9 6 0 2 0 = a a a a W = " c ( l , x ) = r ( p r X )" & "pr x <. Tr" & "pr x = true"; 10 6 0 20 3 cu 4cu W

11

= a a a = "c(l , x ) ; r ( p r X >" & "pr x

6 0

20

3cu

a > Tr" & "pr x

4cu

= false";

during

GENERALIZED NETS AND CCWUTER SCIENCE

r

l1 I 7| I 1 I fl II

= 4

1 99 wW 12

1

W

W

305

10 10 Ww 13

12

13

1 I false 9 I

true

where - 1W

W

;

12

13 a a W = "x = pr x ". 13 1 10 The presented GN model of a DFS does not describe all sible situations in ally during are made and

the pos­

a full-scale system, which is rarely needed. Usu­

the time when

a DFS is designed,

some restrictions are imposed.

particular assumptions This is done to prevent

the possibility of eventual indefinite states in the system, to avoid deadlocks and other undesired ten achieved after

effects.

a reasonable

A predictful behaviour is of­

compromise between the complexity of

a model and the clarity of its operation. In this particular case some potential events are ignored, such as conmunication failures,

invalid client requests or

These situations are easily solvable,

access rights.

but the model would be unreaso­

nably complicated and will not be adequate for its present purpose. Further research work in the field of DFS design will probably bring forth

other problems, and methods for

thor's belief is that a precise GN model of

their solution.

The au­

any distributed file sys­

tem could suggest helpful ideas about its design and development.

306

CHAPTER 5

§ 5. 2: GENERALIZED NET MODELS OF DATA TRANSFER IN INDUSTRIAL LOCAL AREA NETWORKS Alexander Savov

§ 5.2.1

Introduction This

paper by means of the Generalized Nets (GNs) presents the

results of a simulation study carried out by an Network (ILAN).

Industrial Local Area

The results are very close to the designed specifica­

tions, The ILAN is composed of three layers: physical, data

link, and

application [1]. The data link layer

is built on a polling mechanism

because of its deterministic behavior.

All stations are polled either

centrally

by a supervisor or decentrally by a token

[£]. There are two types of stations:

passing protocol

initiators and responders. Ini­

tiators provide the following data transfer services [3]: (a) SDA - send data with acknowledgement, (b) SDH - send data with no acknowledgement, (c) RDR - request data with reply. The initiators connected to the bus cooperate by using the sha­ red channel. The right to use the channel is named a token. A token is passed

from a station to a station in a cyclic way,

thus defining a

logical ring. The services SDA and RDR

are a couple of frames

sent via the

medium: the frame - request - passes from the initiator to the responder, the frame - answer - passes from the responder to the initiator. The service SDN has only one frame request passed from the ini­ tiator to the responder. The stations

included in the ILAN

are Programmable

Devices

(PDs) - components of manufacturing systems such as programmable logic controllers, robots, process controllers, personal hers.

computers, and ot­

Their base functions are calculating the control

programme and

updating inputs and outputs. The control programme works cyclicly. The

GENERALIZED NETS AND COMPUTER SCIENCE

cycle is called

307

a Treatment Cycle (TC). Tie PD takes inputs -when the

TC starts. This is done in order to avoid hazards due to value changes during a cycle.

For the same reason data exchange is impossible

when

the TC runs. Two models of ILAN exist: (a) model 1 - only initiators are included in a logical ring, (b) model 2 - all stations are included in a logical ring. Model 1 behaviour 1. The initiator receives

the toKen and sends

the frame - request to

the responder, 2. The receipt of this frame causes

the responder to wait

the

TC to

end, to compose and transmit the frame- answer. 3. the receipt of this frame causes the initiator to pass the token to another initiator. Model 2 behaviour 1. The initiator receives the token,

sends the frame - request to the

responder, and passes the token to the next initiator. 2. The receipt

of this frame causes the responder to wait

for the TC

to end, to compose the frame-answer, and to wait for the token. 3. Tne responder receives the token and transmits the frame - answer.

§ 5. 2. 2 Generalized

net model

By means of GNs we shall describe

a GN-model

(see Fig. 5. 3)

of data transfer in ILAN. N is the number of stations included in the ILAN. I is the number of initiators included in the ILAN. Lmin is the minimum length of the data field

that can be tras-

ferred by one service. Lmax

is the maximm length

trasferred by one service.

of the field

of data

that can be

308

CHAPTER 5

Fig. 5. 3 CI1 is a TC in the initiator when it composes a frame-request. Cli is a TC in the initiator when it processes a frame-answer. CR

is a TC in the responder when it processes a

frame-request

and proposes a frame-answer. The GN-model consists of two parts. tions of

the ILAN and the second

The first models the func­

(transition r , place 1 and token 5 9

GET'ERALIZED NETS AND COMTJTER SCIENCE

309

a) - the determined protocol on the data link level. The tokens p , . . . ,p 1

in the GN describe

the transfer and pro-

N

cessing of the data in the media and the station. The tokens in the first part of the GN have the following tial characteristics:

ini­

<SA, DA, L, typo, where

(a) SA is the address of the station initiator, the source of the ser­ vice; (b) DA is the address of destination station - responder, the receiver of the service; (c) L is the length of the

data field

that is transferred during the

service; (d) type is the type of service (SDA, RDR, SON [2]). These tokens pass consequently through the places 1 , 1 , 1 , 1 , 1 , 1 , 1 6 3 6 4 6 5 7

1 , 1 , 1, 8 1 2

and 1 . 8

One service is modelled by passing a token through the net. The places model the following: 1 - setting up of the frame-request in the initiator; 1 1 - frame-request's buffer in the initiator; 2 1 - transmission of the frame-request and frame-answer in the media; 6 1 - processing 3

of the

frame-request and setup of the frame-answer's

buffer in the responder; 1 - frame-answer's buffer in the responder; 4 1 - no physical meaning; 5 17 - processing of the frame-answer in the initiator. The time intervals,

when the tokens are in

the different pla­

ces, depict the processing of the data in the stations and the time of transmission in the media

These time intervals are a function of the

310

CHAPTER 5

length of the data-field and depend mostly on U ) the speed

of the microprocessors

and

the hardware in

the data-

transfer section (DMA or other), (b) the data transmiting speed in the media, (c) the frame organization and the data representation. The following equations are used for estimation of the duration Pf a tokens' stay a certain place: (a) for 1 : Tl - the time for setting up the frame-request 1 Tl = Kl+L. tl+RCIl (for the services SDA, SDN), Tl = Kl+R. CI1 (for the service RDR), where Kl - a constant, tl - a time constant depending on the speed of the hardware, R - a random number in the interval [0, 1],
depending on the speed and the method of

the media transmission, S - the number of the service characters in the frame, (c) for 1 : T3 - process time of 3

the frame-request and setting up of

the frame-answer in the responder T3 : K3+L. t3+R. CR;

GENERALIZED NETS AND COMPUTER SCIENCE

311

(d) for 1 : T7 - process time of the frame-answer in the initiator 7 T7 - Kl+L. ti+R. CI2 (for the service RDR) TY = Kl+R. CI2 (for the services SDA, SDN) (e) for 1 : Tfl - time for generation of a request, 6 T8 = R. t4, where t4 is a time constant. The token a in the second part of the net has

a characteristic

"a" which determines the number of the stations that have an access to the media, "a" is changed on every firing of the transition.
= 1 1 11

W 1

1 1 false 6 1

1

1

1 4

3 false

2

false W

W 3

5 false W

4

where

w

: "the time parameter of the token is <. the current time",

w

: "the token has been in place 1 before going to 2

1

2

16" &

"the time parameter of the token is <, the current time",

w = "the token has been in the place 1 before going to 3 3

16" &

312

CHAPTER 5

"the time parameter of the toKen is £ the current time", W

= "the token has been in the place 1 before going to 4 4

16" &

"the time parameter of the token is £ the current time"; 1 6 rr

= 11

1 77

II W W 2 || 5

false false

1 II Wl 3 II

false false

1 II ttrue rue 4 II

false false

1 || ffalse alse 5 II 5

true true

2

where W

= "a = SA for the token", 5 1 1 88 rr

= = 11 33

II W W 77 || 11 11 11

rr = = 11 II W W 44 88 || 11 11 99 rr

= = 11

5 5

9 9

II W W

1 1 1 1

The predicates in the transition conditions for model 2 are si­ milar to the above-mentioned ones with the following difference: 1 1 16 17 6 7 rr = ff aa ll ss ee = 11 II W W 22 22 11 55 I 1 I Wl false 3 I I 1 I

W

4 I 6 I 1 I false 55 I

false

true

GENERALIZED NETS AND COMPUTER SCIENCE

313

where W

= "a = DA for the token". 6 The characteristic

places

functions give

to the toKens

characteristics which are related to the

at different

time-stay of the to­

Kens at these places. The models so sults

described

for the mean time of

are simulated.

a service,

There are certain re­

the optimum traffic

and

the

length of the buffers in the different ILANs. REFERENCES: [1] ISO - Open System Interconnection (DIS-ISO 1*96) [2] ANSI/IEEE

Std.

802.4-1985.

Token passing bus access method and

physical layer specifications. [3] IEC - Process data highway - PROWAY - A, B, C.

314

CHAPTER 5

§ 5. 3: GENERALIZED HET - MODELS OF FFT AND TRANSFUIER SYSTEMS Antonia Dimitrova

Many signals cribed

and

Rossen Fetrov

and phenomena exist in nature which could he des-

and explained by the power analytical apparatus of

formation

the trans-

named after the notorious French mathematician Jean Batiste

Joseph Fourierr. By the Fourier analysis

an arbitrary spatial or time

function can he decomposed into sinusoidal constituents, each of which possesses its own amplitude, represents

frequency, and phase. The transformation

a transition from the time area to the frequency area;

it

is a basis for solving many research problems. In essence, the Fourier transformation performs a spectral analysis of the examined

values. A

basic goal of the implementation of the Fourier transformation is minimizing

the time for its computation. For this purpose a variant for

accelerated

transformation computation

was developed

by Cooley

and

Tukey in 1965, as Fast Fourier Transformation (FFT). By computation of the FFT upon

K discrete readings (reports, samples), the acceleration

which is obtained by the FFT implementation compared with the standard Fourier transformation is K/Log K, r

where

r is the radix of the FFT.

The radix of the FFT could be 2, 4, fl, 16, etc. variant

The most

widely used

is the one with radix 2. In this case the number of

cessed readings algorithm is

is an exact power of 2.

its implementation in m

the pro-

A typical feature of the FFT consequent phases,

Log K. The Fourier transformation is applied to 2

where

m =

both real and complex

input signals. The output signal is always complex. All the intermediate

results

set

up grave requirements on computational environments in which

is implement. of

the FFT

cells is

are complex values too.

That is why with

These peculiarities

the problem for

a maximum load

of parallel

topical and very important.

parallel working

A detailed

(features) FFT

implementation computational

analysis of the FFT

GENERALIZED NETS AND COMPUTER SCIENCE

315

algorithm shows nonlinear dependency of the algorithm

acceleration as

a function of

the number of parallel processors. The aim of the built

up

to

model

is

set up

the number of

parallel

working

which will accelerate the FFT algorithm in a maximum way. tational environment

an INKS IMS T80O

processes

As a compu­

transputer network is chosen.

The number of transputers building up the 16. The basic features of these processors,

transputer network is 2 concerning

to

the built mo­

del, are as follows: (a) four

serial communication channels for connection with processors

of the same type, (b) independency of the data transfer on the computational process; (c) built in device for floating point arithmetic with

10 1WLOPS per­

formance; (d) tact frequency 20 Mfe. Let us have possibility to include tational network.

p transputers in the compu­

In the common case, it is desirable for

power of 2 due to the

p

to be a

fact that the algorithm is implemented by using

radix 2 in order that a maximum and uniform load of

the computational

network is obtained. In this presentation we have a maximum of 16 com­ putational cells which

are connected under a rule each by each. Beca­

use of the large period for the data transfer, conmensurable with computational

time,

a large number

of computational cells

the

does not

correspond to the great speed. This sets up the considered problem It has been examined that for the considered problem bet of cells to over

16 is not effective and this

increasing the numis

the reason why

such cases not to be considered here. Fig. 5.4 ve

represents a generalized net. Using it we shall

the problem about the number of the necessary computational

to achieve a maximum performance. The data volume is significant

sol­ cells when

this problem is discussed. That is why it is set up as an initial fea-

316

CHAPTER 5

ture of the token in place 1 . By changing this feature, a new initia1 lization of the network is necessary. In addition to the token in pla­ ce 1 , another token in place 1 exists and it has a "maximum time for 1 S problem execution" feature. This "maximum time" depends on the will of the operator. For instance, it can be taken from the given task,

e. g.

some boundary time. The first

token

represents a problem in its initial look: the

i FFT which is to be implemented on K points, over p transputers (p = 2 for 0 s i <, 4) maximum speed.

and it is required to determine which variant After the transition

Z

has the

this problem is split into

N

1 2 subproblems, where N - (p - 1), as on each of the subproblems a serial number

is set

in order that they

be distinguished from each

other.

Each subproblem is given a priority in so that the next token does not pass

the transition which has as the input both the next and previous

tokens.

Fig. 5. 4 The first token possesses the highest priority and the priority is

reduced consequently

to the last token.

The transition

by which

GENERALIZED NETS AND COMPUTER SCIENCE

317

such a situation can arise is Z . 4 In the

transition Z

the features of the tokens

in places 1

5

7

and 1 containing the total subproblem execution time are compared. As 8 a result, in the place 1 the token which has the least execution time 9 will be obtained and the problem will be solved at this moment.

where

Z = <[1 , 1 ), 1 1 3

(1 , 1 1, r , M , v ( l , 1 )>, 2 3 1 1 1 3 1 2

-

r 1

1 I false 1 I I 1 I true 2I

1 3 W l false

where W

= "The number of the firings up to the moment, n < N"; 1

Z = ; 2 2 4 2 2 2 Z 3

= <{1 !, [1 ), r , M , v(l )>; 4 5 3 3 4

where the transition condition predicates of r

and r 1

Z = <(1 , 1 J, (1 , 1 ), r , M , v(l , 1 )> 4 5 6 6 7 4 4 5 6 where 1 6 6 1 I W r - 5 I 2 I 4 1 I false 6 I

1 7 7 W 3 true

where W

= the number of transputers is > 1", 2

= -m ■,

w 3

Z

2 : < i l , l ], [ 1 , 1 , 1 ], r , M , A(1 , 1 )> 5 7 8 8 9 10 5 5 7 8

are "true"; 2

318

CHAPTER 5

where 1 8 r

5

=

1 1 7 i I 1 I 80 I!

w

v W" false

1 9

1 10

w v V"

nw & ivi"

W" & -m'

W

where W -

"The l a s t c h a r a c t e r i s t i c of the token

in place 1 7

i s l e s s than

the last characteristic of the token in place 1 " 8 ' W" = "The transition firings are equal to 0", W " = "The transition firing number is N". All elements of M (1 i i i 5) are 1. i

For every token

a

the

following (consecutively) is valid: Ti (a) = (Log p+l, Log p K

2

Tokens enter with the

1).

2

initial characteristic

"Data volune, k"

and they receive the following ones: $ -> "current number of transition firings"; 3 $ -> "k. T ", where T is the one data item transfer time", 4 t t $

2 -> "k. T . ((Log k) - N + l)/(2. (N-l) )", where T 5 c 2 c r a t i o n computation time; 2

$

6

$

-> "k. (T

+ T )/(2x(N-l) t C

-> "k. T ";

7

t 2 -> " ( n - l ) ".

$ 9

);

is thetoasicope-

GENERALIZED BETS AND COMPUTER SCIENCE

319

§ 5. 4: IMPLEMENTATION OF GENERALIZED NETS FOR SIMULATION CF HQDE-TO-NQDE CQMMUNICATIOU IN A REGULAR STRUCTURE Stefan Stefanov

The case

of node-to-node

commmcation in

a massively linked

network of nodes is discussed. A structure

which

we shall

call

a massively linked

regular

graph (MLRG) is shown in Fig. 5. 5. Each node must have 8 bidirectional links. In case the hardware has only

6 or 1 bidirectional

cluster

like the ones shown in

links each node may Fig. 5.6.

be replaced

In that case

by a

we call

the

structure a massively linked regular cluster graph (MLRCG). Bearing in mind that a) the nodes are very close to one another in space, and, b) the main difficulties

in conmini cation

are

the conflicts between

the messages trying to pass through the same link at the same time, the distance (or the message price) between two nodes N

and N i

be defined as the number of links which the message in order to reach S

may j

passes through

starting from N . j i

The distance (message price) between N i

and N is J

P(N , N ) = max (lv(H ) - v(N ) I, lh(N ) - h(H ) I) , i j i j i j where h(N ) i

v(H ) i

is the relative vertical position of H

is the relative horizontal position.

in the graph and i

Note that this result

is

optimal (even better than could be expected). Let D = IE, W, H, S, NE, HW, SE, SW] be the set of the possible "world" directions. Let us substitute v(N ) - v(N ) = v i j

320

CHAPTER 5

h(N ) - h(N ) = h. i j Then P(v, h) = max(|v|,

|h| ).

The primary direction calculation mechanism (PDCM)

is given in

the table below, where /'"-", if x < 0 sgn(x) -)

"0", if x : 0 , I " + ", if x > 0

d(v, h) € D, "NONE" means that no direction is calculable. ( v , hh) ) I 1 1I sgn(v) sgn(v) I ssgn(h) g n ( h ) I dd(v, 11 0 0

1 00

1 None

II

I 1

00

I

--

IW W

||

1I

00

I

++

I EE

I1

1I

-

I

00

I NN

II

1I

-

I

--

I NW

II

1I

-

I

++

E I NNE

||

1I

++

I

00

I SS

II

1I

++

I

-

I SW SW

I|

1I

++

I

++

E I SSE

II

The secondary direction calculation mechanism type of A (SDCMAI is defined recursively: applicable if h / 0 and v ^ 0 and v / h (if (|v| > Ihl) then d(v, 0) d'A(v, h) -1 \ if (Iv| < |h|( then d(0, h) The secondary direction calculation mechanism type of B (SDGMB) is given by the table below which is based on the PDCM The "nearest possible direction" to d(v, h), where D, is defined as follows:

d'B(v, h) €

GENERALIZED NETS AND COMPUTER SCIENCE

321

I1 d(v, d ( v , h) h ) I d'B(v, d'B(V, h) h ) 1I

1 I

NE

I NNor orEE

I 1

E

I NE NEor orSE SE 1I

1 I

SE

I E or S

II

1 I

I 1

S

I SE or SW 1 I

1 I

SW

I S or W

I 1

W

I SW or NW I

1 I

NW

I WWor orNN

I 1

N

I NW NWor orNE NE 1I

I

1I

The tertiary direction calculation mechanism (TDCM) determines d"(v, h) € D - {d(v, h)I, i.e. , d"(v, h) is any member of D different from d(v, h). In the case of a MLRCG

each node is a cluster and the in-clus­

ter directions are calculated similarly (using tables). Note that if the SDCMA is chosen, instead of the FDCM, the mes­ sage' s price

P does not increase, i. e. an optimal routing is achieved

if the FDCM or the SDCMA, is used. If the SDCMB is chosen instead of the FDCM then P

is increased

If the TDCM is chosen instead of the PDCM then

P

is increased

Tertiary directions are chosen in emergency cases

(hardware or

by 1.

by more than 1.

software malfunctions). A problem that Let

is solved by the simulation is the following.

C be the set of the vertices of the graph,

Cs c C be

the

set of message senders, Cd c C be the set of message destinations, nl - |Cs|, n2 = |Cd|. The time W ; W(C, nl, n2) during which all messages wait to re­ ach their destinations is a topic of importance.

322

CHAPTER 5

The generalized net (GN) representing the case is shown in Fig. 5. 7. The model

consists of 7 transitions

further minimized,

but this is not done to

and 11 places.

It can be

preserve clarity

and the

closeness to physics of the process. Tokens from the initial place L

pass through the trivial trani

sition

T

where they receive their initial location coordinates

cc €

1 Cs, destination coordinates

cd € Cd,

with priority equal to the time

when they enter the transition, Wc and Wg being zeroes and all the ot­ her elements of the characteristic vector blanks. Transition T

stays inactive until all nl tokens gather in 2

Passing through the trivial

T

a token receives

L, 1

its next partial de-

2 stination en e c,

calculated

using the PDCM Then it continues thro­

ugh T and goes either to L (1 £ i £ K, and K is the number of no4 7, i des in the graph) if it is empty, or to L if otherwise. From L 8 7, i tokens go to

L

and from there either to L 9

if cc = cd,

or to

o

L

to 2

continue moving in the net. A token from

L

goes either to 8

tial destination, or, Passing

through

T

L

to retry to reach

its par-

5

if it has tried for more than a new partial destination

X

times, to L , 6

en € C

is calculated

3 using the SDCM. The simulation stops when all tokens gather

in L . o

is the maximum retry count. o * The time-parameters of the net are: T = o , t = 1 , t =oo. The transition conditions are

x

GENERALIZED NETS AND COMPUTER SCIENCE

323

L 1 r

= L

1

I

W

I I I

where W = "TIME > nl"; 1 L 3 r 2

= L I true 1 I L l true 2 I L 4

r 3

- L | true 6 I

L 7,1 r 4

L 7,2

...

L 7,N

L fl

- L I W 3 1 2

W 2

...

W 2

W

L I W 4 -| 1 22

W 2

...

W W 2 2 3

L I W 5 1 2

W 2

...

W 2

3

W

where

W = "C(L "c(L 2

, TIME) = 0" & "Cn = : cC(l (l 7, i

)", 7, i

where "i" i s the ni«l>er m«i4>er of the current place; W

= nW ;

3

2

L 5 r : L I W 5 8 I| 4 4 where W

4

= "W

C

W - iv ; 5 4

< X",

L 6 W 5

3

324

CHAPTER 5

Fig.

5. 7.

GENERALIZED NETS AND COMPUTER SCIENCE

325

L 7, 1 L r r = = 6 6

| true 7, 7, 1 1I I l I L | L | true true 7, 2 7, 2 I I

L

| true 7, N I L O

r =L I W 7 9 1 6

L 2 W 7

where W 6

; "C = C "; c d - nW .

W 7

6 Every toKen has only one (its current) characteristic "
en, Wc, Wg>", where (a) cc are the current coordinates; (b) cd are the destination coordinates; (c) en are the coordinates of the next partial destination; (d) Wg is the global wait time (retry count); (e) Wc is a scratch (current retry count). The characteristic functions related to the places are $

-> "", 1

where (a) cc is the initial location (the message source); (b) cd is the destination; (c) en is undefined; (d) Wc is 0; (e) Dg is 0;

326

$

CHAPTER 5

-> ""; 3

where

"en" is the next partial destination calculated using the PDCM,

all the rest remaining unchanged; $ = ""; 4 where

"en"

is the next partial destination calculated using the SDCM

(A or B), all the rest remaining unchanged; $ = ""; 5 where (a) Wc = Wc + l; "; 9 where cc = cd, all the rest remaining unchanged;

$

-> "*";

2 $ -> "*"; 7, i $ -> "»"; 8 $ -> "<Wg, TIME - nl>", O (this is the final token's characteristic;

TIME

is the current model

time). All the places and arcs have infinite capacities. All the tran­ sitions and places have equal priorities. All transition types are "v".

GENERALIZED HETS AND COMPUTER SCIENCE

327

§ 5. 5: GENERALIZED HET MODELLING OF DISK SUBSYSTEMS Bistra B. NiKolova

Disk Drives (DDs) and the subsystems one of

the most important components

built on their bases, are

of the computer.

They are used

for expansion of the RAH of the computer. The productivity of the com­ puter depends on them An example of a structure scheme shown

in

Fig. 5.8.

It consists of

and H numbers of DDs. The whole

of a disk subsystem

an input/output path

(DS) is (i/o path)

software and hardware, necessary for

the data exchange between the RAH and DDs is represented as a separate functional unit, placed between the RAH and DDs [1]. This unit is mar­ ked as an i/o path. part in

If a certain

the i/o path of

part of

the hardware,

which takes

the concrete subsystem, is occupied by a DD,

the other DD cannot exchange data. Data exchange is realized by direct memory access.

Fig. 5. 8. There are some kinds of architectures

in which in order to in­

crease the parallelity in the functioning of the DS, new i/o paths are added. They are

called "subsystems with a dynamic distribution of the

control between a certain number of i/o paths". They consist of H num­ bers of DDs and connected DDs and

H

numbers of i/o paths,

to every i/o path.

2 i H i H [4]. Every DD is

The control of the i/o operations

with

their parts can be done with the help of as i/o path. The re­

quests to an i/o path form a queue [2,5].

328

CHAPTER 5

Usually, the i/o path is used for but also added, for

of control information.

two buses

are formed

of data,

If a separate path for commands

for data and commands.

positioning at every moment of

the data bus

exchange, not only

the DD

is

The opportunity

is created,

nevertheless

and the i/o path are busy with the data exchange from/to

a DD. The result of an i/o operation is given in Fig. 5. 9, where A - the request goes into the DS (in the queue to the DDs) B - the i/o operation is started C - the positioning of the DDs is completed and the request goes

into

the queue of the i/o path D - the data exchange begins E - the request service ends and the request leaves the DS T - the time during which the request is in the queue to the DDs DQ T - position time DS T - the time, during which the request is in the queue to i/o path CQ T - data exchange time CS

Fig. 5. 9. On the other hand, an i/o operation in the DS consists of positioning, searching and exchanging data. The purpose of the article is to make a model of some DS archi-

GEI>JERALI2ED NETS AND COMPUTER SCIENCE

329

tecture in terms of Generalized Nets (GNs). Some specifications are made: 1) all DDs are equal and

the input requests

are equally

distributed

between them [2, 5]; 2) the fulfilment of the i/o

operations includes the steps from Fig.

5.9 [2,5]; 3) the queues to the DDs and the i/o path are realized by means of the FIFO [2, 3]; 4) every D has its own queue of requests,

while the queue to

the i/o

path is conmon; 5) DD positions the head over the wanted cylinder by itself; 6) the i/o operation starts from a free DD and i/o path; otherwise the request

moves either to the queue of the DD or to the queue of the

i/o path, according to the architecture of the DS; 7) the i/o path

takes

part in

the search for the block on the track

and in the exchange of information between the DDs and the i/o path [33. The requests in the DS are the tokens in the given model. token a goes in the net with an initial characteristic:

Each

"
a NB, LDBL, IOO>" (= x ), where NDS, NCYLN, NB, LDBL are natural numbers 0 and NDS is the number of the DD, 1 i NDS i MaxNDS, where MaxNDS is the ma­ ximum number of the DD, NCYLN is

the

number

of

the cylinder,

MaxNCYLN is the maximum NB is

the number

0 £ NCYLN <, MaxNCYLN,

where

number of the cylinder,

of the block

on the track,

1 4 NB i MaxNB,

where

MaxNB is the maximum number of blocks on the track, LDBL is

the length of the exchange

data

in

the blocks,

1 <, LDBL i

MaxLDBL, where MaxLDBL is the maximum length of the exchange data

330

CHAPTER 5

in the blocks, equal to

the information capacity of the DD in the

blocks, IOO is the code of the i/o operation, 100 = , is

the code of

where READ

the read-operation, WRITE is the code of the wri-

te-operation. A request

service has

been modelled

as a transfer of a token

through the GN described in Fig. 5. 10. The places in the net used for modelling of different architec­ tures are denoted as follows: b - models the preparation of requests, 0 bJ - distributes the requests to the corresponding one of the m-th DDs, 0 b

- a queue to the DD, 1

b

- the DD is free, 2

b' - the DD is not free, 2 b

- the i/o path is free, 3

b' - the i/o path is not free, 3

Fig. 5. 10.

GENERALIZED NETS AND COMPUTER SCIENCE

331

b" - the independent path of commands is free, 3 b"' - at least one of the N numbers of the i/o path is free, 3 b"" - none of the N numbers of the i/o path is free, 3 b

- start of an i/o operation, 4

b

- independent positioning of the DD over the desired cylinder, 5

b

- a queue of the i/o path, 6

b

- a free i/o path and search for a blocK, 7

b' - the i/o path is busy and search is not possible, 7 b" - all the N numbers of the i/o path are busy, 7 b

- exchange between DDs and RAM. 8 First, we shall discuss a DS which consists of a DD and

path,

as is shown in Fig. 5. 10. Each token

characteristic enters

the net.

an i/o

cc with the above initial

The request goes

in the queue of the

DDs. This queue is processed by means of a FIFO-discipline. The GN following

modelling

this architecture

components and

parameters.

is characterized

b 1 b I true 0 I I b I true 2I b

I true 3I b 2

r 1

= b I W 1 1 1

the

It contains 6 transitions with

conditions:

r = 0

by

b' 2 w 2

332

CHAPTER 5

where W 1

= "c(b , TIME) = 0"; 2

2

= 1W ; 1

W

(the function

c(l, TIME)

determines the number of tokens in place

at the time-moment TIME); b 3 3 r

=b 2

I W | 2 | 3

b' 3 3 W 4

where W 3

= "c(b , TIME) = 0"; 3

4

= nw ; 3

W

b 4 r

= b 3

I true 3I b 5

r = b | true 4 4I b 7 r

= b 5

| true 5I b 8

r

= b 6

I true 7I

The characteristic function is defined as follows: ^"<TIME, TIME - 6 (a)>", if a goes into b for the first time K 2 $

-> ■

2

a "<TIME, TIME - pr x >", otherwise V. 1 cu

1

GENERALIZED NETS AND COMPUTER SCIENCE

333

a where x is the current characteristic of token a; cu a $ -> "<TIME, TIME - pr x >"; 3 1 cu a a $ -> "h (pr x , pr x )", where the function h determines the ti4 1 2 cu 2 cu-1 1 me for waiting in place b ; 4 a a $ -> "h (pr x , pr x )", where the function h determines the time 5 2 20 4 0 2 for positioning; a a $ -> "h (pr x , pr x )", 7 3 20 4 0

where the function h determines the time 3

for searching for the blocK; a a $ -> "h (pr x , pr x )", 8 4 30 4 0

where the function h determines the time 4

for exchange between the DD and the RAM. Second, we shal1 discuss a DS which consists of an independent command path, an i/o path, and M DDs, as it is shown in Fig. 5. 11. The independent command path controls the search and exchange of in­ formation. Each toKen a with the above initial characteristic enters the net. We accept that in practice the independent command path is always free. The queue to the DDs and the i/o path, as above, are pro­ cessed by means of a FIFO-discipline. The GN modelling this architecture is characterized by compo­ nents, which are similar to the above ones. The transition condition r is 0 b b ... b 0, 1 0,2 0, M r =b I W 0 0 | 1 where

W

... 1

W 1

334

CHAPTER 5

GENERALIZED NETS AND COMPUTER SCIENCE

W

335

a = "the number of place is equal to pr x ". i 1 0 The transition conditions r

are i, i

r

= b 1, i b

b b

1, 1, i i | true 0, i |

| true 2, i I

for 1 i i i VL The transition conditions r

r , r ,. . . , r 2, 1 2, 2 2, M

are similar to

with exactly the same notation. 1 The transition conditions r

and r 3

have the forms: 4

b" 3 b r

= 3

| true 2, 1 I I b | true 2,2 I

b

I true 2, M I

b 4, 1 r = b" I W 4 3 1 1

b

. . . 4, 2

W

b 4, M

. . . 1

W 1

Noting that the capacity of place b" is 1. 3 The transition conditions r

r , r ,..., r are similar to 5,1 5,2 5, M

with exactly the same notation. 5 The transition conditions r

and r 6

have the forms: 7

336

CHAPTER 5

b 6 b r

= 6

I I true true 5, 11 II 5, I I bb II true true 5, 22 II 5,

b

I I true true 5,M 5,M II

b' b' 7

II true true II

b b 77 rr ss bb II W W 77 66 11 55

b' b' 77 W W 6 6

where W

= "c(b , TIME) = 0"; 5

W

7

= nW ; 6 5 The transition condition r

is the same as the one in the first

8 GN. The characteristic functions in both GNs coincide exactly in nota­ tion. Finally,

we shall discuss a DS which consists of DDs in number

M and i/o paths in number N, as it is shown on Fig. 5. 12.

The discip­

line to the DD and the i/o path, as above, is FIFO. The differences between the second and the third net are in the forms of transitions conditions r , r and r . They have the forms: 0 3 7 b b .. .. .. b b b b 0, 0, 0, 0, 11 0, 22 0, M M r = b I W WJ ... W 00 0 1 11 1 1 bb"" "" II 33 1

W 11

1

W 1

...

W 1

GENERALIZED NETS AND COMPUTER SCIENCE

337

338

CHAPTER 5

b"' 3

bb"" "" 3

I 2, 1 I I b I 2, 22 II 2,

W

W 8

b

W

b r

= 3

7 W

W 77

I 2,M I

7

88

W 8

where W 7

= "c(b"', TIME) : 0"; 3

w = -m ; 8

7 b

b 7, 1

r

= b 7

I W 6 1 9

. . . 7, 2

W 9

b 7, N

b" 7

. . . W 9

W 10

where W

= "This i/o path is free"; 9

W

= "There is no free i/o path". 10 The transition conditions

r , r 8, 1 8, 2

r

are similar to 8, N

r with exactly the same notation. 8 The function $ is similar to the one from the second GN. The theory of the GNs makes it possible to create a cannon for­ mal description of DS. The concept GN allows

all phases of an

dependency between them when i/o requests be shown.

i/o operation

in the DS

and the

are processed to

It also allows all specialities of the separate architectu­

res to be shown.

GENERALIZED NETS AND COMPUTER SCIENCE

339

REFERENCES: [1] J. Major, Processor, i/o path and DASD configuration capacity, IBM System Journal, Vol. 20, No.

1981. 63-85.

[2] I. Donev, A research over the productivity of disk subsystems, [3] D. Brown, R. EiQsen and C. Thorn, Channel and direct access device architecture, IBM System Journal, Vol. 1 No. 3, 1972. 186-199. [4-] R. Wilson,

Designers rescue

superminicomputers from

i/o bottle­

neck, Computer Design, Vol. 26, No. 18, 1987, 61-71. [5] I. Donev,

A research over

the productivity

of disk

subsystems.

Proc. of Symp. "Electronic and computing technics and technology", May 1987, Sofia, (in Bulgarian). [6] I. Donev, L. Michov and N. Sinyagina,

A research over the organi­

zation of highly-productive disk subsystems with stratification of information. Automation Control, Systems, Sofia (in Bulgarian).

Computing Technic

and Automated

340

CHAPTER 5

§ 5. 6: SOFTWARE TOOLS FOR GENERALIZED HETS Rumen Christov

§ 5.6.1

and

Stojan Hibov

Introduction During the last ten years concurrent processes

ve been developed.

Too many big

researches ha­

and complex systems do not allow

an

analytic study. Petri Hets (PNs) and their extensions and modificati­ ons are a powerful theoretical tool for the study of the properties of complex concurrent systems. Too many software tools for implementation of the PNs and some of

their modifications

work of scientists easier.

are produced to make

the

These tools contribute to further develop­

ment and extension of the practical applications

of PNs and their mo­

difications [1], Each extention of

the FHs is

designed to solve problems in a

determined area. Howadays a trend to generalization and comparison mathematical

models

But there is no

of processes

from different areas

of

is observed.

compatibility between the FHs' extentions.

It

makes

the comparison of the processes' models very difficult. Generalized Hets

(GHs) were defined in 1982.

They are a po­

werful mathematical tool for describing and studing the properties

of

complex concurrent systems. These nets are a continuation of the trend to make tools for modelling of parallel processes. GHs are an extension of the PNs. It is proved, that all exten­ sions

and modifications of -the PNs are particular cases of

fact proves

that each process

GHs. This

which may be described and studied

by

FHs or with their extentions may be described and studied by GHs. This means

that GNs

are a universal

language for describing and studying

the processes from different scientific and practical areas. This fact proves the neccesity

of software tools for

the im­

plementation of GHs. They are developed by the group, working over GNs and the tools are named STGNs. The STGHs include:

GENERALIZED NETS AND CCMTJTER SCIENCE

34-1

(a) presentation of GN-based hierarchical models of processes; (b) computer tool support, with which it is possible to create and si­ mulate hierarchical GN; (c) tools for analysing results of a GN-based model simulation.

§ 5. 6. 2 GTfs - a short

description

GNs are a complex

mathematical object.

Many of

their compo­

nents are presented in the other FNs extentions. However, the GNs are not an aggregate

of the components of other types of PNs.

All compo­

nents of GNs have a specific content. GNs have a static and a dynamic structure, temporal components and a global memory. Static structure The

static structure includes the net transitions, net places

and arcs which connect the places with the transitions. The information necessary for each transition description is: (a) a transition

identifier,

which is

a unique name in

the net for

them; (b) a transition name (only for user applications); (c) priority towards other transitions in the GN; (d) temporal components of the transition: time of the next activation and duration of this active status; (e) type of the transition; (f) indexed matrix expressing the transition conditions; (g) a list of the input places for the transitions; (h) a list of the output places for the transitions. The transition type, condition matrix and the functions which assign temporal transition

components are included in the dynamic net

structure. The transition type contains the necessary conditions

for the

342

CHAPTER 5

transition activation.

In the GNs' theory

the transition type

has a

specific structure, but in software tools it may be an arbitrary logi­ cal expression, which may depend on arbitrary events in the net at the same or a certain previous step of the model simulation. The information necessary for each place description is (a) place identifier, which is a unique name in the net for than; (b) place name (only for user applications); (c) priority towards other places in the GN; (d) place capacity - maximum number of tokens which may be in the pla­ ce in one step from the net simulation; (e) tokens which

are in

the place

before

the GN-roodel

simulation

starts. A place may be (a) a GN input, if it is

a member only of

the input places' list for

any net transition; (b) a GN output, if it is a member only of the output places' list for any net transition; (c) internal to the net, if it is a member of for any net transition and is a member of

the input places'

list

the output places' list

for another (eventually the same) net transition. From

each input place for each transition in the GN,

each of its outputs are defined. For each arc a predicate from

A capacity

arcs

to

is assigned to each arc.

the transition condition matrix corres­

ponds. Dynamic structure Components of the dynamic structure are tokens, types of tran­ sitions,

conditions of

the transitions and temporal functions of the

transitions. The information necessary for each token description is (a) a token identifier, which is a unique name in the net for them;

GENERALIZED NETS AND COffUTER SCIENCE

34-3

(b) a token name (only for user applications); (c) priority towards other tokens in the GN; (d) the time when the token enters the GN; (e) a list of the net input

places,

through which

a token may enter

the GN. At the moment

when the net starts to simulate, the tokens may

be in any place in the net or may be waiting to enter the net.

In the

beginning, the tokens may have a list of any initial characteristics. The token's way

through

the net depends

When the tokens are going in the net,

on

these characteristics.

to the lists of characteristics

new characteristics are appended The characteristics are the greatest difference between GNs and other kinds of PNs. dividuality

The tokens have an in­

during the whole time of net simulation.

This fact means

that many of the properties of the concurrent processes may be descri­ bed

by characteristics.

helps models

the user

It simplifies

the static net structure and

to understand the model of the process.

are more compact

The GN-based

than models based on other kinds of PNs.

It

allows more detailed models of the studied processes to be made and to get more complex results. Token characteristics are useful but their software implementa­ tion is very difficult.

They may not be formalized in their full the­

oretical definition. Software implementation is the implementation

of

a class of GNs. However, this class is enough to describe and to study real

and theoretical processes

and to use these software

tools for

practical applications. The characteristics nents of

have two aspects.

First,

they are compo­

the net PSI-function. It is a definition of the way for cal­

culating token characteristic values. Second, they are the list of va­ lues, which a certain token gains when it comes into

a certain place.

The following types of characteristic values are supported: - a whole number;

344

CHAPTER 5

- a real number; - logical values TRUE and FALSE; - a string; - a list, containing an arbitrary combination

of

values of

the above types; - a matrix of whole or real numbers. Conditions of transitions The conditions of

net transitions

are

an indexed matrix

of

predicates (see App. 2). The matrix elements are indexed by transition input and output place

identifiers.

Each predicate of these matrices

corresponds to one of the transition arcs. The values of

the predica­

tes are calculated at each step of the model simulation for each token in the corresponding input place. Predicates may sions which return the values TRUE or FALSE. ons between all net attributes.

be arbitrary expres­

They may include relati­

A very important limitation

is

that

the predicates cannot depend on further events in the net. Type of transition The type of transition is a logical expression. It assigns the necessary set

of

tokens at the transition input

places for

regular

work of the GN. If the type of transition returns TRUE, then the tran­ sition has been activated. Global memory The global memory includes; (a) initial token characteristic values; (b) characteristic functions,

which act on

each token at

the places

corresponding to them; (c) the number of

the characteristic

token is in the net.

values which

are kept when the

GENERALIZED NETS AND COMPUTER SCIENCE

Wien the token is at a net's characteristic

values.

net. It allows the

output place,

345

it owns a list

of

The list corresponds to the token path in the

process to be studied by the history of its compo-

nents represented by tokens in the GN model. Hierarchical models based on GNs GNs allow one to create hierarchical models of the studied pro­ cesses. It is useful for their complex study. of subnets.

A GN may contain

a set

In the primary GN these subnets are represented by places

or transitions. If the subnet has only one input and one output, it is represented by a place. outputs,

If the subnet has

several inputs and several

it is represented by a transition.

If a token comes

into a

certain subnet input, then the subnet simulation starts. The simulati­ on of the primary net and its

subnets is synchronized by time towards

an absolute time scale. The subnets can use information for the events in the primary net and other subnets tion.

As a result the tokens gain

during the time of their simula­

the result of the subnet action as

a characteristic. As a further development be

other PNs extentions,

it is provided

that

the subnets may

as Predicative/Transition Nets

or Coloured

Petri Nets for example or Neural Networks. This fact means: - models, based on different PNs'

extentions may be compa­

red by their actions and results during the time of their simulation; - models, based model;

already built, may be included as parts of

- if a Neural Networks

based model

based model, then each Neural Network

is a subnet of the GNs

may be taught and it may define

the action of the primary net,

§ 5. 6. 3. Tools for <3V

a GN

representation

The software tools for implementation of GNs consist of:

346

CHATTER 5

- a language for description of GNs based models; - a graphic editor which allows to construct, edit and update GNs based models; - a tool for simulation of GNs based models; - a tools for analysis and representation of the results of the simulation of the model as histogranms, piecharts, etc. Language for GN description The models

based on GNs

are represented by language for

a GN

description (LGND). The LGND is designed and implemented specially for the description of the Its

reserved words

describing ment and

GNs. It

is a high-level programming language.

are notions from the GNs' theory.

the GN structure

Besides tools,

the LGND contains tools for

evaluation of predicates

the assign­

and standard functions for all at­

tributes of the net; current time, time, when a selected transition is active, number of tokens in the selected place and many others. Construction and editing of GNs The

editor in the STGNs allows the user to construct

and edit

complex hierarchical GNs. The user works with a GN on a computer, sup­ porting high resolution graphic screen and a mouse. The graphic repre­ sentation of the GN uses theoretically nents. of

Each editing operation

the computer.

defined symbols for its compo­

is immediately reflected on the screen

The user can increase or decrease some parts of the

GN image, so that he can have a global view over the model, created by him The user has

an access to each net element.

attributes by a menu system He may move or delete a

He may change its selected element

or may observe the subnet, which this element represents, ment is a place or a transition.

if the ele­

GENERALIZED NETS AND COMPUTER SCIENCE

The STGNs

recognizes the structure of GNs.

347

It makes

a syntax

(by the GN theory) analysis to the GN, created by the user and reports him about errors. GN simulator The STGNs includes a GN simulator. This is a tool for simulati­ on of the GN, described by the LGND. The simulation corresponds to the conmon algorithm Seme variances are allowed Certain tokens can split and join in certain net places.

If the splitting

is allowed,

token may go from one input to several output places, in the net as many new tokens are as

their

children place,

born with

when a

it splits and

the same characteristics

parent as the number of the output places is. When all the of one parent (tokens from one split level) are in a certain

they can join to become one token.

tics will not be a list.

The new token characteris­

They are a net with nodes

characteristic of

different tokens at the same time moments. In a certain practical app­ lication, a c annum cat ion of different tokens is allowed. The user

follows the simulation on the screen step by step. He

can see the token's movement through the GN. The simulation can be manual or automatic. In manual simulation each GN step may be defined by the user. In automatic simulation, only the first step is defined by the user. the system

Each next step is generated by

(GN model). In both cases, results of the traversed

steps

are generated. The user can define the appearance

of

the results of

the GN

model simulation. He can use histogramms and tables for representation of the change in the characteristic selected by him for kens.

He can

for one

follow the relations between different

different to­ characteristics

or several tokens. He can follow which transition was not ac­

tive in a selected time interval, kens and other features.

which place was not reached by

to­

3*8

CHAPTER 5

§ 5. 6. 4.

Conclusions

The STGNs are a powerful tool the properties of

using

the GNs for

the study of

the practical and theoretical concurrent processes.

Ttieir software support and development has to be continued. Currently, the STGNs is used for description and simulation of

the GNs based mo­

dels, described in the this book. The next step is to make these tools more user-friendly and to extend

the class of GNs,

which can be

described and simulated

with

them REFERENCE: [1 ] K. Jensen

and

R. Shapiro,

A tool package,

supporting to

use of

Colored Petri Nets, Petri Net Newsletter, Vol. 35 (1988), 22-36.

Appendix 1: REMARKS ON GENERALIZED NETS

In this

Appendix

we shall give

the basic definitions related

with the concept "generalized net" following the contents of

the bock

"Generalized Hets".

* a

§ 1. On the concept "generalized

«

net"

We shall give a formal definition of

the concepts

"transition

of a GET and "a GH". Every transition is described by a seven-tuple: Z = , 1 2 where a) LJ

and

L"

are finite, non-empty sets of places (the transition's

input and output places respectively); J

1.1 they are L'= (1 , 1'

1 b) t

2

for the transition in Fig.

1'J and L" = II", ]",...,]"].

1

m

2

n

is the current time-moment of the processes's firing; 1

c) t is the current value of the duration of its activity; 2 d) r is the transition's condition determining the

349

tokens which shall

350

APPENDIX 1

Fig. 1. 1. transfer from the

transition's inputs to

its outputs;

it has the

form of an index matrix (see App. 2): 1" . . . 1 1'

r

-

1

V i : 1' m

(i, j)

1" . . . 1" j n

I | I I I I I I I

denotes the element

r i,j (r

- predicates) i. j i,

( l i i i m l i j i n ) which corresponds to the

i-th input and

j-th output places; these elements are predicates and when the truth value of the

(i, j)-th

element is true,

the token from

place can be transferred to j-th output place; otherwise, possible;

i-th input it is not

REMARKS ON THE GENERALIZED NETS

351

e) M is an index matrix of the capacities of transition's arcs: I"

1

1' M

I' m f) D

.

.

1" . . .

j

1"

n

1 1 1

: 1' i

=

.

is an object having

1 m 1 i,j ; I 1 (m i 0 - natural lumbers) 1 i, j 1 | |1 U U 1 i j ( n| a form similar to a Boolean expression.

In

it the variables are all symbols which mark the transition's inputs nemes, and the Bollean operations "A" and

"v" determine the

fol-

lowing conditions: Ml

i

.1 1

,...,1 ) - every place 1 , 1 i i i i 2 u i 2

1 i

must

contain

u

at least one token, v(l

i

, 1 , . .., 1 i i 2 u

1

) - in all places 1 , 1 i i 1 2

1 i u

tain at least one token, where (1 , 1 , . . . , 1 i i i 1 2 u

must contain

) c L'.

The ordered four-tuple O

K

E = <, , , <X, f, b>> A L 1 2 K K is called a Generalized Net (GN), if: a) A is a set of transitions; b) n A

is a function giving the

priorities of the transitions,

i. e. ,

TI : A -> N, where N = 10, 1, 2, . . . ) U too); A c) TI L

is a function

L -> N,

giving the

priorities of the places,

where L = pr A U pr A, and 1 2

i. e. , TT : L

pr X is the i-th projection of i

the n-diroensional set, where n € N, n i i and 1 <, k i n (obviously,

352

APPENDIX 1

L is the set of all GN-places); d) c is a function giving the capacities of the places, i. e. , c:L-> N; e) f is a function which calculates the truth values of the predicates of the transition's conditions

(for the GN described here, let the

function f have the value "false" or "true", i. e. , a value from the set

(0, 1).

In § 3. 1,

we shall

describe other

types of nets in

which this function will have a value in the interval [0, 1] or in the set [0, i ] x [0, 1 ] ); f) © is a function giving the 1

next time-moment

when a given transi-

* © (t) = t', where t, t' € [T, T + t ] 1

tion can be activated, i. e. ,

and t £ t'. The value of this function is

calculated at the moment

when the transition ends its functioning.

Below we shall discuss a

special representation of this function; g) © 2

is a function giving the

duration of the activity

of a

given

K

transition, i.e., © (t> = V , where t € [T, T + t ] and t' £ 0. The 2 value of this function is calculated at the moment when the transi­ tion starts its functioning.

It has

another representation as the

above one; h) K is the set of the GN's tokens. In some cases, it is convenient to consider this set in the form K = U I 1€Q where K

K , 1

is the set of tokens which enter the net from place 1, and 1

I Q i) TT K

is the set of all net's input places; is a function giving the

priorities of the tokens,

i. e. ,

TI : K

K -> N; j) 9 K

is a function giving the time-moment when a given token can en-

REMARKS CM THE GENERALIZED NETS

353

ter in the net, i. e. , 6 (a) = t, where K a 6 K,

*

t € [T, T + t ] ; K) T

is the time-moment when the GN starts functioning.

This moment

is determined about a fixed (global) time-scale; o 1) t is an elementary time-step, related to the fixed scale;

(global) time-

M

m) t

is a duration of the functioning of the net;

n) X

is the set of all initial characteristics

which can receive the

tokens when they enter the net; o) $

is a

characteristic function which gives new characteristic to

every token when it makes a given transition.

a transfer from input to output place of

As some of

the above functions

this function

can be represented in another form, shown below. p) b

is a function giving the maximum number of characteristics which

can receive a given token, i. e. , b: K -> N.

If for a certain token

a, b(a) = 1, the token will enter the net with an initial characte­ ristic (as a zero-characteristic). After this, it will receive only the characteristic of the previous. When b(a) = a>, the token a will receive all possible characteristics. When b(cc) = k < co, except its zero-characteristic, the token a will keep the last

k

as its cha­

racteristics (previous characteristics will be "forgotten"). Hence, in general, every token a has b(a) + 1 characteristics. We must note that this definition similarly to

the ordinary ON

definition is not fully formalized, because if we fully formalize transition

conditions and the characteristic functions

the

of the GNs, a

smaller class of GNs will be obtained. It is convenient

to assume

that the functions

f, 6 , © i 2

and

354

APPENDIX 1

$ have the following forms, also: IAI f = U f , where f calculates the truth-values of i-1 i i the i-th

GN transition

the predicates of

(as we shall see in

§3.1,

they

cannot have values "true" and "false"; they can have fuzzy or intuitionistic fuzzy transition

conditions

(see can

App. 2)

be

values;

calculated

different

by

different

IAI i i 6 = U 9 , where 9 calculates the next time-moment of the 1 i= l 1 1

activati-

ways);

on of the i-th GN transition; IAI i i Q - U 0 , where 9 2 i=l 2 2

calculates the

duration of the active state

of

the i-th GN transition; ILI $ = U $ , where $ calculates the i= l i i

tokens'

characteristics,

which

they will receive in the i-th GN place. A given GN may not have some of the components. Then the places of such components will be marked by "*". The GNs with such form gene­ rate a special class of GNs, which we shall describe in § 2.1, The static part of a given GN is determined by the set pr

{i

i i

A, where for a given n-dimensional set X (n £ 2) i, 2, 6, 7 K pr X - n pr X i ,i i j=l i 1 2 k j

in, l i j i k , j

of

the elements of

i i- i JJ J"

for j' t j"),

a GN is determined by input and output

i.e. the static part

transition places,

dex matrix of the arcs and by the type of the transition. cal character of the net comes from the GN's tokens and ons conditions

(pr A), 5

by in­

The dynami­ the transiti­

the temporal character comes from

the compo-

REMARKS ON THE GENERALIZED NETS o * nents T, t , t and from the elements of the set pr components $, X and

b

355

A. 3,4

Finally, the

play the part of the GN's memory.

The different functions are also related to these four GN com­ ponents: functions TT , n , c to the static structure; f, IT to the dyA L K namical elements; 9 , Q and 6 to the temporal components. 1 2 K One very important restriction for

the GNs

is the

following.

The transition condition predicates cannot be related to future events for the GN. X K

K

§ 2. Reduced GNs For two subsets £' and E"

of E - the class

of all GNs

define (see also definitions of the relations n, n , s, * E' t- £" iff the functioning and

x

in

let us § 5:

*

the results of the work of every ele­

ment of E" can be described by some element of E'; E' i- E" iff "HE* H E"), E' H E" iff <E' i- E") & <£" i- E'), E' H E" MI

T(S' H

E")-

A given GN may not have some of the components. Then the places of such components will be marKed by "*". The GNs with such form gene­ rate a special class of GNs. Let us define the set of the GN's components o * (A, TI , IT , c, f, e , e , K, TT , e , T, t , t , x, $, b) u A L 1 2 K K (A / 1 £ i £ 7}, i where A = pr A (1 i i <■ 7), i. e. , A € I L \ L", t , t , r, M, ol i i i 1 2 Q

:

and l e t Y € Q.

and

356

APPENDIX 1

Y By E we shall denote the class of those GNs

which do not have

the Y-component. Y Y Obviously, £ i- £ for every Y €ft.Elements of the class £

we

shall call "reduced GNs". Evidently A A

£ because

A 1

=E

2

=E

there does not exist

component A)

K

=2=1*,

a GN without

graphical structure

and without tokens (the component K);

(the

without input and

output places of its transitions (components A

and A respective). 1 2 Y Hie class £ will be called "a 1-class of reduced GNs" for eve-

ry Y € ft. If Y , Y 1 2

e 2 for s i i,

Y

then

Y , Y 1 2 E

Y s will

s

be called an "s-class of reduced GNs", Y , Y ,..., Y 1 2 s We shall note also that, if £ * 0,

A , A , Kj, then I

and

Y € (A,

Y , Y ,. . . , Y , Y 1 2 s £ i- ft.

2 O A , A , A , A ,TT ,11 , c, 0 , 0 , n ,© , T, t 3 4 6 7 A L 1 2 K K

* ,t

, X, b

THEOREM 2. 2. 1: £

H £. x *

§ 3, Conservative

extensions

K

of the OVs

Eight different types of GN-extensions are defined and for each of them it is proved that it is a conservative extension. First type intuitionistic fuzzy GNs In the definition of component f

the concept

GN

it was noticed

that the

(the function which checks the truth values of the predi-

REMARKS ON THE GENERALIZED NETS

357

cates) can give different type values. In the first case they are ele­ ments of the set ("false", "true") (or (O, 1J); in the second - it has values

in the interval [O, 1] and then the GN which contains such

component is called a Fuzzy GN; in the third case - it has

f-

values

in the set [O, 1] x [O, 1] and the corresponding GN is called Intuitionistic fuzzy GN, The above two nets are denoted in brief by FGN1 IFGN1 respectively. FGNs" and

We shall call

"first type IFGNs"

these first types GNs

(for the concept

and

"first type

"intuitionistic fuzzy

set" see App. 3). It is obvious

that every FGN1 is a IFGNi. Every GN of this ty­

pe has the form E = <i
IT

K

o * , e >, , <X, $, b>>, K

where the elements of the set A (the IPGN's transitions) as the

are the same

GN's transitions. All other components, without the components

f and § are also the same. The function

$ now gives to every token as

current characteristic two values: the first coincides with characteristic in the sense of the GN's;

the token

the second value is an orde­

red tuple with real numbers, every one of which is an

element of

the

set [0, 1 ]. They are equal to the truth values of the predicate of the transition condition between the place, from where starts the transfer of

the corresponding

token and the place,

where this transfer ends.

The function f calculates these two values of the corresponding predi­ cate r i, J ff(r (r where p(r

)

> = <jj(r <jj(r ), ), T(r T(r )>, )>, i,j j i, i iiJ iJ i,i,J J

i s t h e degree o f t r u t h of the p r e d i c a t e

i,j i s i t s degree of f a l s e ,

r

, i, j

and (cf.

App. 3)

p(r ) + T(r ) <, 1. i, j i, j

T(r

) i, j

358

APFEHDIX 1

Second type intuitionistic fuzzy GHs Here we sbal1 give a complete formal definition of

the concept

"second type intuitionistic fuzzy GHs" (IPGH2), In this net the tokens are some "quantities" which move inside the net. The values of the transition condition's predicates can

be in­

tuitionistic fuzzing s, i.e. they can have degrees of truth and of fal­ seness (see App. 3). The IPGN2 has the form o « E = <, , , <X, $>>, A L I E K K where A is the set of GH-fonn dex

(cf. § i)

matrix

H

the net's transitions

which have the

ordinary

and there exists only one difference: here the in­

contains as elements real numbers - capacities

of the

transition's arcs. The functions in the

first component are similar to

those for

IFGN1 and they satisfy the same conditions. The essential

difference between IFGH2S and the other

the set K and the functions related to it. some "quantities", which have as (elements of the set tics. The function 8

X)

How the

initial characteristics some

and which do not receive

gives the time-moment when a

GHs

is

elements of K are "type"

other characteris­ given token

will

X

enter the net as in the ordinary GHs. The temporal components are also as in the other GH-types. The function $ has new meaning. How it is related to the places and they receive

by this

function characteristics (the quantities of

the tokens from the different times in the

corresponding places).

As

in GHs, it can be extended: it can also give other data for the model­ led process (e. g. the time-moments for

the entry of the

"quantities"

in the places. The definition of the second type FGH (FGH2) is analogous. For IFGH2 the following are valid:

REMARKS ON THE GENERALIZED NETS

THEOREM 3. 2. 1: For every

359

IFGN2 there exists a GN which represents it.

THEOREM 3, 3. 2: For every GN there exists a IFGN2 which represents it. Colour GNs They have: (a) toRens coloured in n different ways (1 i n <. <x>), (b) arcs coloured in R different ways (1 i R i co). Arcs of

various colours will spring off from every place

not more than R in nuntoer), going into every place GN's

transition will

and also arcs of various colours will

(but not nore than have the form of

(1 i i < p) and 1 < R " <. R (1 i j i s). j

Fig. 3. 1.

R

in nuntoer).

Fig. 3. 1,

where

(but be

Thus every i i R '£ R i

360

APPENDIX 1

(1 i i i p) and i i K " i K (1 £ j <. 5), j The token

tinted in colour which is narked with number

brief "colour i"> should only pass through a coloured arc, of which is an element of the set I .

A n arc numbered

j

i

(in

the number should host

i only tokens tinted in the colour with a number which is an element the set J . Obviously: j

of

n In U I I = n , IiU= l I iI = n , i= l i k k I . I u u J J i i = = k k . J=i J=i J J

The condition of the above CGN transition (which is an arbitra­ ry one) has the form (1", q" ) . . . (1", q» ) . . . (1", q" ) . . . (1", q" > 1 1,1 1 1, k" s s, 1 s s, k" 1 s
(i','
> i

i 1

1, K" lI l,k' r = . 1 I : I ( J ' I q" I ti'.V > i P P, P, K' K' I I P I p

rr qq" , q» ii P Ji (i i

P

<. k', l i i

T

T

<, k", i <. i <. p, l <, j £ s) j

where r is a predicate related to the q' -th output arc of q' , q" i,P i, P j. T the input (for the given transition) place \J and to the q" - th i j, T put arc of the output place 1". j transition has the form

Analogously,

the

in-

M-component of the

REMARKS ON THE GENERALIZED NETS

361

d", q" ) . . • d", q" I . . • U", d", q" ) .■ .■ .■ (1", U". q" ) 11,1 1 1, K" k" s s, 1 s s, k" 1 s d', U \ cf ) I 1 1, 1 I IV, IV,'f "q"

l M

-IV,

P P

cf

)

p, i 1 p,

(1J, V P

)I

l,k' I i i I

m qJ , q" i, p P j, J, TT

|

(l s sp p ii k\ l <, <, TT i. k", 1 1 <, <, i is s p, p, l l ii jj ii s) s) (l k\ l i. k", i j i j

I I I I

)> II

P,K' | P I

where m i O sire natural numbers. q' , q" i.P J, T The net so constructed has a set of tokens

K

of the form

K =

n U K , where K is the set of tokens which are coloured in i-th coloi=l i i ur. viously, every ordinary GN is a CGN for which n = k =1. GNs with interval time for activation The transition firing time-moment of this extension of is substituted by an interval.

the GNs

In that way the transitions in the new

net acquire the form: Z - , 2 K

where d - [ f , t"] and T s t' i t" < T + t . When

the tokens

in the

input places of Z are already enough to satisfy the transition type

D

and TIME € a, the transition Z will be fired. Let the GN E be a net which has at least one transition of the form described above. For this the function © is extended to function 1

362

APPENDIX 1

9 , which to every time-interval d = [V, t"] 1 1 1

juxtaposes a new timeK

interval [t\ t"j 2 2

for which (T i)

t" i t' i t" i T + t , 1 2 2

i.e. ,

the

next time-interval for firing of the transition Z will come not earli­ er than the moment when this transition ends its former firing. GNs with a complex structure The transition forms of GNs are graphically similar to those of the E-nets. For the Petri nets the graphical structure is not the same as for GNs.

It is proved

that if a constructed GN which

than one arc entering each of its places

allows more

and more than one arc coming

out of each of its places, then this extension of the GNs will also be conservative and there it is called "a GN with a complex structure". GNs with global memory In the modelling

of real processes it

is appropriate

to keep

data (while the GN is functioning) or to determine the values of ferent parameters related to these processes.

dif­

Thus, the definition of

the concept GN can be extended with the addition of a new component B, which we shall call "global memory". The component

B can be seen as a

list of such functions that will change their values in the process of GN functioning.

Let a GN with component B be called "a GN with global

memory" (GNGM). The formal definition of this object is: *

E

O K

= <, , , A

L

1 2

K

K

>, where all components are as in § 1,1 and the component B is as descri­ bed above.

REMARKS O N THE GENERALIZED NETS

363

GNs with optimization components Some GN's models correspond to real processes

whose functions

do not stop and which determine standard (given before) the process's continuation. components

solutions for

F o r these G N s it is necessary to add such

that secure their non-stop functioning.

For this aim,

we

shall construct a new extension of the ordinary GNs - GNs with additi­ onal clocks (GNACs). Itiey have transitions with the form: Z - < L \ L", t , t , r, t , r', M, Q>, 1 2 3 where the components conmon with the ordinary transitions are the same and a) r' is a (0, 1)-index matrix with the form 1" . . . 1

1" . . . j

1" n

1' I 1 1 I I : 1' i

r' =

I r' I i, j I I (r € (0, 11) I i. j I I (1 S I $ m, 1 < J ,j i n)

: 1' m where L' = IV,

1 b) t

1 J ) and L" = 11", 1"

1'

2

is the maximal

1

m

,

1"),

2

n

time-duration for a check

of the truth

of the

3 predicates of the transition condition

r. In the case that time is

increased the checking of the predicates of

r shall be reduced and

their respective values from the matrix r' will be assigned. Obviously, t 3 the predicate's

s t , but if 2

truth can

t 3

> t 2

the process of checking of

be stopped before

the time necessary

for

364

APPENDIX 1

this tume-moment. If A is the set of the GNAC's transitions the GNAC has the form o E

=

<
n , n , c, f, e , 0 , e , a>, A

L

1 2


3

«

n , e >, , K

K

<X, $, §', b>>, where a) 6

is a function

giving the duration of the transition's condition

3 K

checking, i.e., 0 (t) = t', 3

with t € [T, T + t ]

and t' i O.

The

value of this function is calculated at the moment when the transi­ tion starts its functioning and,

after this, at every time-moment,

when the condition's predicates are calculated and the current step of the token's

transfers are finished

and there exists

more such

steps. This function determines the values of the transition compo­ nent t , and if © (TIME) = t' and 3 nent t

TIME + t' > t

3 does not receive new value.

+ t , 1

the compo-

2

3 b) G

is a function which to the predicates of r Juxtaposes (0, 1)-va­

lues by an advance determined way.

Its calculations are made inxne-

diatelly after each checking of the values of the function © . 3 c) #'

is a function which

tokens which

determines the new characteristics

enter the transition's

output places

of the

after applying

the specifics of the new nets mechamism,

» x

§ 4-. GNs and other

it

objects

In this chapter,

it is proved that the functioning of

the re­

sults of the work of the different types of modifications of the Petri nets and of the finite automata and the Turing machines can be sented by GNs.

A GN which makes the same for each

repre­

ordinary Petri net

REMARKS ON THE GENERALIZED NETS

365

is constructed. •

§ 5. Algebraic

aspect

of the theory

it

of OVs

For the two t r a n s i t i o n s Z and Z we s h a l l 1 2 Z = Z i f f ( v i : 1 i i £ 7 ) ( p r Z = pr Z ); 1 2 i 1 i 2 Z c Z 1 2

iff

define

(Vi: 1 £ i £ 2)<pr Z c pr Z ) & i 1 i 2 (Vi: 3 £ i £ 4 ) ( p r Z = pr Z ) & i 1 i 2 (Vi: 5 £ i £ 6>(pr Z c i l l

pr Z ) & (pr Z c l 2 i l 2

pr Z ), 12

where c

is a relation of inclusion over index matrices and if 1 A = (X , L , la I], B = [ K , L , 1 1 i, j 2 2 A C B iff 1

c

(K 1

lb I], then i,j

c K ) & (L C L ) & (Vi€K >(Vj€L ) ( a 2 1 2 1 l i ,

= b ) j i , j

i s a r e l a t i o n of i n c l u s i o n over Boolean e x p r e s s i o n s and, f o r two of 2 these expressions a and b a c

b iff it is getting the expression

a after removing

the argu-

2 merits of b, which are not arguments of a and the logical operations, associated to them Over the transitions Z 1),

i i i i i i i = < L , L , t , t , r , M , D > i 1 2 1 2

we shall define three operations.

It is necessary for

(i=l,

the follo­

wing to be valid: if place 1 € pr Z n pr Z and 1 € pr Z , then 1 € 1 i 2 i s 3-i pr Z f o r l £ i £ 2 , l £ s £ 2 . These o p e r a t i o n s are 3-s 3-i

366

APPENDIX 1

a) a union

(the necessary conditions

for this operation are

(j = 1,2), and (as if 1 e pr Z , this is s i pr

not possible),

1 2 t - t j J that

1€

Z for 1 i x i 2, 1 <. s <, 2): 3-s 3-i 1

Z V Z 1 2

=
2 1 U L , L 1 1 2

2 1 1 1 U L , t , t , r 2 1 2

2 1 + r , M

2 1 2 + M , V(D , D >>,

b> an i n t e r s e c t i o n (with the above c o n d i t i o n s ) : 1

Z HZ 1 2

=
2 1 D L , L 1 1 2

2 1 1 1 n L , t , t , r 2 1 2

2 1 x r , M

2 xM,

A(Q

1 2 , Q )>.

c) a composition (with the above c o n d i t i o n and w i t h t h e c o n d i t i o n 1 2 L nL 1 1

= L 2

Z

=
1 2 f l L = (*): 2 1

oZ 1 2

2 1 2 1 2 1 1 U | l - L |, L U ( L - L ), t , t , r 1 1 2 2 2 1 1 2

1 2 1 2 or,M oM,

1 -2 v(D , D )>, where D results from D after removing all its arguments, whose iden1

tifications are elements of the set L 1 It is possible that L Z

o o n Z = Z , where Z 1 2

2 n L 1 1

2 n L. 2 1

- f> and

L

1 2 n L = . In this case 2 2

is the empty transition

(for

some other coro-

ponents as M, r, Q will be degenerated). "Hie operations described below are unique in the Petri net the­ ory.

Tliey can be transfered on practically

nets

(obviously with some modifications

the corresponding

nets).

all other types

related to

These operations are

of Petri

the structure of

very useful

for con­

structing GN models of real processes. Before introducing the different GN's operations, we shall for-

REMARKS ON THE GENERALIZED NETS

367

mil ate same conditions which the arguments of these operations (diffe­ rent GNs) must satisfy. We shall assume that

if a place

takes part

simultaneously in

two nets, then in both locations the place has (a) equal capacities, (b) the same characteristic functions (c) equal possibility to accept tokens net

from K,

(which is a model of some process)

ments of the set

K' c K

could pass

i.e. if

in the first

the tokens which are ele­

through the examined place,

while in the second net tokens which are elements of the set

K" c

K could pass through this place, then K' = K", (d) equal values of the priority functions, (e> one and the same numeration/notation. Similarly,

we shall expect that if two places are respectively

an input one and an output one for two transitions in both nets, then (a) the capacities of the connecting arcs are equal, (b) the time for passing from the input to the output place

should be

one and the same, (c) they should be in the same consequence, i. e. the place which is an input

(output) one in the first of the transitions

should be the

same in the other transition too. Let E

and E 1

be two GNs and let for i = 1, 2: 2

i i i i E = <
i i , 6 , 9 >, 1 2

i i , i K K

o * , i i i

.

A union will be called the object: 1 E

U E

1

=

<
2 2 6 U6>, 2 2

U A ,

1

n

2

1


2

1

U T I . T I

A

A

2

1

U T I . C

L

2 U c , f

1

2 Uf,

L

1 2 1 2 TI U T T , 6 U 6 > , K K K K

1

2

6

UO,

1 <min(T , T ), 1 2

1

O o GCD(t , t ), 1 2

368

APPENDIX 1

» o o o + t . t /GCD(t / G C D ( t , t ) - m i n ( T , T )>>, ))>, i i 1 2 1 2

max (T iixi2 i <X 1

U X , $ U $ , b U b >>, 2 1 2 1 2

where A

U A 1

= 2

2 U ( Z / ( Z € A ) & (V Z' i=l i 2 U fZ/(3Z' i=l

€ A

) (Z D Z'

o = Z )) U

3-i

o € A )(3Z" € A ) (Z' D Z" ?! Z > & (Z = Z' U Z " > ) . i 3-i

A composition of the above nets will be called the object: (E , if T + t < T 1 1 2 2 1 E o E = 1 1 2 \ x E , i f T <, T + t ^ 3 1 2 2 where 1

E

=<
1
2

UK, 1 2

A

2

1

un

, i

A

L

1 2 TI V T HI , K K

2

1

un,c

2

uc,f

1

2 1

uf,

e

2 1

1 2 © U © e >, K K

* o o o t . t //GCD(t G C D ( t , t ) - T )>, i i 1 2 1

o o
<X 1

U X , $ U $ , b 2 1 2

1

2

ue,e

L

ue>, 1

2

2

max (T + liiS2 i U b ». 1 2

Before introducing the next operation over GNs, we shall defiene a local operator O - an

operators for place identification, which juxI

taposes to every given GN E and to every given places 1 € Q I E O d a new GN E 0(E,

1 , 1 ) = <<(A 1 2 [
L",

(
t , t , r, 1 2

L",

t

, t , r, 1 2

M, D > / ( 1 " € L")

M, D > / 1 "

and 1 € 2

E l " l | D

&
U"!)

U

(l')>

REMARKS ON THE GENERALIZED NETS

& (
L",

TT , e >,

K

K

t , t , r, 1 2

o * ,

M, D> € A ) ] > ,

369

71 , TI , c, A L

f,

9 , 9 >, 1 2


<x, $, b>>.

I t can be e a s i l y seen, t h a t the s e t s f/1" € L"! 1 2

and

[ < L \ L", t , t , r, M, D>/(1" € L") & (L" = (L" - [1")) U ( l ' l & 1 2 ( € A)) 1 2 are one-element s e t s (of t r a n s i t i o n s ) . We s h a l l , lows.

extend the operator O over two s u b s e t s of L, as f o l -

If L' = ( 1 ' , 1 and 1J * 1', 1' * i J i 0(E, L \

I 1' 1') C Q and L" = fl", 1" 2 n E 1 2 1", 1" n ' (1 H , j S n ) , then J i J

L") = 0 ( . . . 0 ( 0 ( E ,

1', 1

1"), 1', 1 2

I")--, 2

O 1"! c Q n E

1', n

1">n

Obviously, the new object will be a (34 too. The object E» for which there exist sets h' - fl*,. 1' 1 2 I O c Q and L" = (1", 1" 1") c Q and Ex = O'E, L \ L") E 1 2 n E fixed GN E for which L = L , and for every i E an oriented

1'] n for some

(1 s i <. n) there exists

path from a place 1' to a place 1" i i

in

E,

we shall call

iteration of the (24 E For every two GNs

E

and

E

1

different types of relations are 2

defined in [«], but we shall give here only these which

are used

in

the book. Let us define for a given GN

E

two sets, one of tokens K and

one of the initial token's characteristics X:

370

APPENDIX 1 «

K(E) = K(E) = la/(ce ( a / ( a €€ K> K) & & (6 (6 (a) (a) <
with which

the token a can start its transfer in the GN E. Obviously, X(E) c U X(a>. a€K For a given token a € K and a given initial characteristic x 6 X(E) let

aa x >, if if a a € € K(E) K(E) and and x x € € X(cc| X(a| ( , )) fin fin E(a, < E(a, x) x) -. =<

rr (
II x>

f

a where x fin

,, otherwise otherwise

a a aa a a


>, if a e K(E) and x € X(a)

,

, otherwise

, i f a e K(E) and x € X(a) 1 2 fin

is the final characteristic of the token a in the GN E. ,

, otherwise

Let us define E (a, x) = E fa, xi = . Below we shall introduce four relations between the GNs: E 1

c E 2

iff

(K (E ) c K (E )) & (X (E ) c X (E )> & (Va € K (E )> 1 1 2 2 1 1 2 2 1 1 (VX € X (E ) ) ( E (a, X) = E (a, 1 1 1 2

E c E 1 x 2

iff

x||

(K (E ) C c KK (E )) & (X (E > ) c X (E )) & (Va € K (E )) 1 1 2 2 1 1 2 2 1 1 (VX € X (E ) ) ( 3 i , i 1 1 1 2

i : 1 <, i < . . . < i <, f i n ) s 1 s

(E (a, xl = pr E (a, x) ), 1 i , i ,. . . , i 2 1 2 s E s E 1 2

iff

(E 1

c E ) & (E c E ), 2 2 1

REMARKS ON 1HE GENERALIZED NETS

E

s E 1 * 2

iff

(E

c 1

371

E | & (E c E ). 2 2 x 1

x

Obviously, the relations c

and K

are stronger than the rela-

The first GN-definition (in § 1. )

in some sense is a deductive

tions c and z respectively.

one.

Below

we shall introduce

a second GN-definition.

In the above

sense, the new definition will be of an inductive type. Let

Z

be a

given object,

having the

grapical structure

in

Fig. 1. 1 and the following components: a) L' - a set of places which we shall call input places, b) L" - a set of places which we shall call output places, c) t

- a time -moment about some fixed

time-scale (with an elementary

1 o time-step t >, d) t

- a real number which corresponds to the length of a time-inter2

val in the above mentioned time-scale, e) r

- an index matrix (see App. 1) having the form 1" . . . 1" . . . 1" 1 j n 1' 1 r

=

1' i : 1J m

(i, j)-th

element of

I I I I l I I I l

r i,j (r

- predirates) i, J

(i i i i m, 1 S. j S n)

which correspond to the

output places, these elements being predicates, f) M - an index matrix having the form

i-th

injwt and j-th

372

APPENDIX 1

1" .. .. .. 1 1' 1'

M = M

1" .. .. .. 1" jj n

II 1 II

1' 1' i i : : 1' rm m

I I ID m l l i,j i,j II I (m i O I 0 -- natural natural lumbers) numbers) II i,i, jj I iI (l£i£m,l£j*n) I ( l £ i £ m , l £ j * n )

g) Q is an object having a form similar to an Boolean expression. variables are exactly the symbols which marK the names; the Boolean operations

"A" and

"v"

Its

Z's input places'

determine the analogo­

us conditions from § 1 and 1)A(1

, ] ,...,1 ) - every one of the places 1 , 1 i i i i i 1 2 u 1 2 contain at least one token,

2) v(l

,1 i 1

,..., 1 ) - all the places 1 , 1 ,. . . ,1 i i i i i 2 u 1 2 u

at least one token, where

1 i

must u

must contain

(1 , 1 ,...,1 ) c L'. i i i 1 2 u

The object described above will be called a transition. The inductive definition of the concept GN is as follows: (1) The object which has the form of a transition

is called a GN,

if

to it are added a) TT - a function giving a natural number, being the priority of the A transition, b) n - a function giving the priorities of the transition's places, L c) c - a function giving the capacities of the transition's places, d) f - a function

which calculates the truth values of the predicates

of the index matrix r; e) ©

- a function giving 1

the next tune-moment

when the

given tran-

REMARKS ON THE GENERALIZED NETS

373

* sition can be activated, i.e. © (t) - t', where t, t' e [T, T+t ] 1 and t i t ' . The value of this function is calculated

at

the mo­

of

a given

ment when the transition ends its functioning; f) © 2

a function

giving the duration of

the activity

transition, i. e. , 6 (t) = t\ where t e [T, T + t ) and 2 The value of this function is calculated

at the moment

t' z 0.

when the

transition starts its functioning; g) K - a set of tokens; h) ii - a function giving the priorities of the tokens (IT : K -> N); K K i) © - a function giving the time-moment when a given token can enter K the net, i. e. © (a) = t, where K a e K, K

t 6 [T, T + t ]; j) T - a time-moment

when the GN starts functioning.

This moment is

determined on a fixed time-scale and the first value of t = T; 1 o k) t - an elementary time-step related to the fixed time-scale; K

1) t

- a duration of the functioning of the net;

m) X - a set of all initial characteristics which can

receive the to­

kens when they enter the net; n) § - a characteristic

function

which gives

new characteristic

to

every token when it makes a transfer from input to output place

of

a given transition; o> b - a function

giving the maximal number of characteristics

which

can receive a given token (i. e. b: K -> N) transfering in the net. (2) If E, E

and E 1

a) E

U E 1

are GNs, then 2

is a GN, 2

APPENDIX I

374 b) E o E is a GN, 1

2

c) Ex is a GN. THEOREM 5, 3, t: The two definitions of the concept "GN" are equivalent. K

» J 6. Topological

»

aspect of the theory of GNs

The graphical structure of a given GN is a two-coloured graph. Thus the study of such a structure and checking of the properties of a GN following from these structures are important for the theory of the GNs.

Some operators which assign to every GN one- or two-coloured

graph and some other operators, which juxtapose to every GN one natu­ ral

number corresponding to the complexity of the net with respect to

a certain criterion, are defined, The class £ is ordered by relations, generated from the above two types of operators, and some topological properties of £ are studied. x K

j 7. Logical

*

aspect of the theory of GNs

Logical operators which are similar to the modal operators "ne­ cessity"

and "possibility"

are defined and some of their basic pro­

perties are given. * *

x

§ 8. Operator aspect of the theory of GNs The operator aspect has an important place the GNs.

in the theory of

Six types of operators are defined in its framework.

Every

operator assigns to a given GN a new GN, which has some preliminaryly, introduced properties. Particular types of operators are: global (one

REMARKS ON THE GENERALIZED NETS

which

transforms globally

transforms

some

some transition

375

GN's components),

components

local

of a given GN),

(one which

hierarchical

(one which substitutes a place or a transition for some part of

a gi­

ven GN, or makes reversal), reducing (one which removes some of the GN components), extending (one which transforms a given GN to the corres­ ponding extending GN)

and dynamical (one which determines

rent strategies for the tokens' transfer

in the net).

the diffe­

The basic task

of this aspect of the theory of the GNs is a research on the connecti­ ons and influence between the separate operators.

* § 9. Other extensions of the 0Vs By analogy with the universal Turing machine, versal GN (UGN) as a GN E

the object Uni-

for which the following is valid: K

(V E € £)(E c is intriduced.

E ),

It is noted that at the moment of writing of [K] (NOV.

1990) , the question for the existence of an UGN is open. Let Q be a some fixed time-scale. For T' e Si we define 2 = I E / E E J SprprE tprprEST'!. T' 1 3 3 3 K

We shall call the GN E

for which the following is valid, K

(V E € £

)(E c

T'

E ),

it

a T'-UGN about the time-scale Q. Obviously, the global time-components of O

E

=

<
A

, n , c, f, e , e >, L


K

1 2

, e >,


K

, t >, <x, §, b » ,

K

it

satisfy the inequality T + t

ST'.

THEOREM 9. 1. 1: For every time-moment T' € Q about the fixed time-scale

376

APPENDIX 1

it

Q, there exists a T' -UGN E

for the class £ T'

Let £ *

= IE / E e £ & pr pr E = x & pr pr E < 00), 1 3 3 3

i. e. , every GN of J x

has finite time for its functioning.

Let E x

be a

X

GN similar to E

but without the first global temporal component. Then

we shall define GN E x

as a universal GN (UGN) for the class £ , i. e. , x x

IVK6I |(Ec E >. X

K

THEOREM 9. 1. 2: There exists a UGN for the class £ . M

Let E be some GN. Let K be a set of tokens of E, and X be a set of initial

characteristics for these tokens of K.

The object

"Self-

Modifying GN" (SMSN) is constructed. x Let K

be a set of tokens which will be called

The defining of this set is possible in the

"control

framework

tokens".

of the GN the-

x ory.

The tokens of K

can be interpreted as GN-tokens

with initial

K

characteristics

(elements of the set X ) which contain the identifier

"control token"

(this is also possible).

frame of a given GN places

(in which

characteristic

E

is that

there will be

The second addition in the

the characteristic functions of transferred control tokens)

some

give as

identifiers and parameters of some types of operators,

defined over the GN.

This is also not in contradiction with the defi­

nition of GNs. Finally (really, the new): when a certain control token gets as characteristic the identifier of some operator with parameters former

or current characteristics of

this or

another control token,

then at this time-moment over the GN E the corresponding operator will be realized. Obviously, not-every operator can be applied to the net

in the

REMARKS ON THE GENERALIZED NETS

377

time-interval when it functions. Thus, we can define the SMGN as a GN, for which there exists a mechanism p for realization of the over it.

This mechanism functions in the following way:

moment when a certain control token of a given ristic the identifier

of a certain

operators

at the tiroe-

GN gets as a characte­

(added) operator and the

values

of its parameters, the mechanism p realizes this operator. The basic aims in [») are: 1. to show the operators which can be realized by p; 2. to show the relations between GNs and SMGNs.

and £

If £ is the class of all SMGNs, then obviously J c J SMQN SMQN H £, because every GN is a SMGN with an empty set of con-

SMGN trol tokens. THEOREM 9. 2. 1: For every SMGN, there exists a GN which represents it. The SMGNs are the basic aim of the operator aspect of

the the­

ory of GNs. x

* § 10. Methodological

aspect

of

»

the theory

of GNs

The b a s i c ways for c o n s t r u c t i n g o f a GN of a given r e a l process and same ways for u s i n g and modifying of already c o n s t r u c t e d GN-models are d i s c u s s e d . X *

§ 11. Open

X

problems

There are many unsolved

or unforroulated problems in the theory

of GNs. Here we shall introduce some of these problems which are rela­ ted to the text of the book. 5. To research the different classes of reduced GNs without two or mo­ re components, and to show the relations between these classes.

378

APPENDIX 1

7. To prove that the constructed

reduced GN in § 2. 2 from the book on

GNs (from § 1. 1.2) has the least possible number

of different GN's

components, or to give a counter example of this. 8. What other conservative extensions of the GNs can be defined? 9. To construct GNs which are universal for the classes (sets) of the different

Petri net modifications,

by analogy

with the

GNs from

§ 4. 3. 12. To construct an algorithm which for every set of transition phical structures

(obviously they must satisfy

some

gra­

conditions)

and for every given GN, construct a GN for which: - all its transitions have

the forms, which are elements

of the

given set, - both nets have equal tokens and as a result of their

with equal

initial characteristics

functioning equal tokens receive equal

final characteristics. 19. To prove or disprove the existence of a universal GN for £. 20. To construct other extensions of the GNs and to prove

or disprove

their conservativness. 21. Chapter 10 is a superficial attempt to give the basics of

the m e ­

thodology of work with GNs. To continue and enrich the research in this area. 22. A list of the basic applications to this moment (Nov. 1990) is gi­ ven in Chapter 0. To extend the area of the GN's applications.

Appendix 2: GENERALIZED INDEX MATRICES

A definition of "generalized index matrix" (GIB) or "index mat­ rix" is given in [1-3]. Here we snail introduce only the results which are related to this basic text. Let bers.

I be a fixed index set and H be the set of all real num­

A GIH with generalized index sets K and L

(K, L c I) will be

called the object: 1 1

k

1

k 2

k m

1

1 2

|a i kK , 1l

1

1

a k ,l

1

. . .

1 n

. . .

2

a k , 1

I n

la a . . . I k ,1 k ,1 I 2 1 2 2 I I . .

a k ,1 2 n

I a a I k ,1 k ,1 | m 1 m 2

a k ,1 m n

< =

[X, L,

.

fa ]]) k ,l i j

),

where X = Ik , k , . . . , k 1, L = 11 , 1 , . . . , 1 !, for l ! i ( n an

1

2

1 2

m

for 1 i j i n: a

n

€ K. Let H be the set of all GIHs with index k , 1 X, L i j sets X. and L, and let H = U H . K, Lcl

K, L

379

APPENDIX 2:

380

In [31, for the GIMs A = [K, L,

(a

k ,1 i j

1], B = [P, Q, (b

a r e defined ordinary matrix o p e r a t i o n s "addition" on"

t ,v u w

/a / k ,1

"miltiplicati-

! ] . where

, if t

- k

u

k

1

b

v "i

t , u w

a

, if t

p ,q r s

t + b

K ,1 i j

p ,q r s

v0

t ,v u w

-

a

k,l i j

, if t

. b

P,q r s

t ,v u w

u

= k

€L-Qor

j

€ K - P and v = 1 e L w j

i

€ P - K and v

r

= p

i

r

6 K DP

w

€Q-Lor = q

and

e Q

s v

= 1 w

= q j s

na

= k i

u

The b a s i c p r o p e r t i e s of

= p r

e K D P and v

=1 w

= q € j s

L n Q; fc

t

u

v

!],

where

w

if t

1 b 1 p ,q r s E a .b 1 =p 6LTP k , 1 p ,q j r i j r s

,°

=1

1], where

, for t

(v. t ,V u w

= p

u

w

= p € P and v = q r w s

e L

A . B = [K U (P-L), Q U (L-P),

c

u

€ K and v

i

, otherwise

A x B = [K n P, L n Q, fc

c

!)

and a l s o the f o l l o w i n g o p e r a t i o n s

A + B = [K U P, L U Q, fc

c

and

p ,q r s

if t

if t

u

u

u

= k

= p

i

r

€ K and v

w

=1

e P - L and v

w

j

€ L - P

= q

s

e Q

= K € K and v = q € Q i w s

otherwise the s e t M w i t h t h e above o p e r a t i o n s are

GENERALIZED INDEX MATRICES

381

studied in [3]. There some GIMs measures are introduced (for GIMs de­ terminants do not exist). Ihe given mathematical

apparatus may be

with elements from the sets (0, li,

applied to

the

GIMs

[0, 1), or from the class of all

predicates, etc. In the first two cases, the operations " + " and ". " in R will be substituted by

"max"

and

"min" respectively,

and in the

third case - by the operations "v" and "*". REFERENCES: [1] Atanassov K.

Conditions in

Generalized

nets,

Proc. of the XIII

Spring Conf. of the Union of Bulg. Math. , Sunny Beach, April 1964, 219-226. [2] Atanassov K. Dynamical elements in the Generalized nets,

AMSE Re­

view Vol. 1 (1985), No. 4, 1-9. [3] Atanassov K. Generalized index matrices, Comptes rendus de demie Bulgare des Sciences, Vol.+0, 1987, No. 11, 15-18.

l'Aca-

Appendix 3 : INTUITIONISTIC FUZZY SETS

Following [1,2] we shall give short remarks on

the intuitioni-

stic fuzzy sets (IFSs) and other intuitionistic fuzzy objects

related

to them. Let a set E be fixed. An IFS A«

in E is an object having

the

form a A

: |<1, |l |I), 1 <X)>/X€E),

A where

the functions

A

y (x) : E -> [0, 1] A

fine the degree of membership

and

t (x) : E -> [0, 1] deA

and the degree of non-membership of the

element x€E to the set A, which is a subset of

E,

respectively,

and

for every x€E 0 i \i <x] + T (x) i 1. A A Obviously, every ordinary fuzzy set has the form |<x, v (x), 1-p <x)>/x€E]. A A 11

For simplicity, below we shall write A instead of A . For every two IFSs A and B, the following relations tions are valid: A c B iff (VX€E) <JJ (x) <, \i (x) 8 T <x) i T (x)); A B A B A D B iff B c A;

382

and opera­

INIUITIONISTIC FUZZY SETS

383

A = B iff (Vx€E)(p (x| = p (x) & T (x) = Y (x)) A B A B A = (<x, T (x), p (x)>/xeE); A A A D B = (<x, minlp (X), p (x>), max(T (x), A B A

t

(x))>/xeE);

A U B = (<x, max(p (x), p ( x ) ) , min(T (x), A B A

t (x))>/x€E); B

B

A + B = (<x, p (x)+p ( x ) - p (x). p (x), T (X). T (x)>/x€E); A B A B A B A . B = [<x, p (x). p (x), A B

T (x)+T ( X ) - T (x). T (x(>/xeE); A B A B

A - B = A fl B. The concept IFS is the basis for the definition of the concepts Intuitionistic Fuzzy Propositional Calculus

(IFPC)

and such extensi­

ons as IF predicative calculus, IF modal logic, etc. (see [3-5]). To each proposition (in the classical meaning) its truth value: truth

denoted by 1, or falsum by 0.

one can

assign

In the

case of

fuzzy logics, this truth value is a real number in the interval and can

be called "truth degree" of a particular proposition.

case of

IFPC is

added one more value - "falsum

[0, 1] In the

degree" - which will

be in the interval [0, 1] as well. Thus one assigns to

the proposition

p two real numbers p(p) and T(p); moreover the following constraint is valid: p(p> + T(P) i

i.

Let this be done by an evaluation function V defined so that V(P) = . Hence the function V: S — >

[0, 1] x [0, 1] gives the truth and

falsuni degrees from the class of all propositions and it

can be defi­

ned so that it assigns to the logical truth T: V(T> = <1, 0>, and to the logical falsum F: V(F) = <0, 1>. Ibe negation np of the proposition p will be defined through

384-

APPENDIX 3:

V H P ) =
P(p)>.

When T(P) = 1 - p, 1 - P(P)>, for np we get V(np) = <1 - p(p>, p(p)>, which coincides with the result, e. g. , from [6]. Depending on the way of defining, of the operation

"3"

can be

obtained by different variants of IFPC. One of them is as follows: V(p & q) = <min(p(p), p(q)>, max
T(q))>,

V(p * q> = <max(p(p), p(q)), min(T(p),

T,

V(p 3 q) = <max(T(p), p(q)), min(p(p>,

r(q))>.

A given propositional form A propositional form; if

(c. f.

[7):

each proposition is a

A is a propositional form, then iA is a propo­

sitional form; if A and B are propositional forms, A D B are propositional forms)

then A & B, A * B,

will be called a tautology,

iff

V(A) = <1, 0>, for all valuation functions V,

and an intuitionistic fuzzy tautology,

iff "if V(A) = , then a i b". There exist some forms of an intuitionistic fuzzy

Modus Ponens

(see [3,5]). The definition for the quantifiers is as follows: V(VxA) = <min p(A>, max T ( A ) > x x and V(3xA) = <max p(A), min T(A)>. x x For the proposition p for which V(p> = , in [4] are defined the following operators: V(Qp) = ,

INTUITIONISTIC FUZZY SETS

385

V(Op) = , vdiich are analogous to the nodal operators "necessity" and "possibili­ ty". REFERENCES: [1] K. Atanassov,

Intuitionistic fuzzy sets,

Fuzzy Sets and Systems,

Vol. 20 (1986), No. 1, 87-96. [2] K. Atanassov,

More on intuitionistic fuzzy sets.

Fuzzy

Sets and

Systems, 33, 1989, No. 1, 37-46. [3] K. Atanassov,

Two variants

of intuitonistic fuzzy

propositional

calculus. Preprint IM-JFAIS-5-88, Sofia, 1988. [4] K. Atanassov,

Two variants of

intuitionistic fuzzy

modal logic.

Preprint IM-MFAIS-3-89, Sofia, 1989. [5] K. Atanassov, G. Gargov, Intuitionistic fuzzy logic.

Compt. rend.

Acad. bulg. Sci. , Tome 43, N. 3, 1990, 9-12. [6] A

Kaufmann,

Introduction a la theorie des sous-ensembles

flous,

Paris, Mas son, 1977. [7] Mendelson E. , Introduction to mathematical logic, D. Van Most rand, 1964.

Princeton,

NJ:

LIST QF AUTHORS

Alexander Georgiev - Institute for Microsystems, Sofia Alexander Savov -

Institute for Microsystems, Sofia

Antoaneta Eirova - High Economical Institute, Sofia Antonia Dimitrova - Institute for Microsystems, Sofia Bistra Nikolova - Institute for Microsystems Borjana Jordanova - Faculty of Mathematics, El. Ochridsky Univ. , Sofia Evgeni Tzolov - Institute for Microsystems, Sofia Ewgeni Dimitrov - High Transport School, Sofia Ilya Eazalarsky - Institute for Microsystems, Sofia Ivan Hristozov - Institute for Microsystems, Sofia Joseph Sorsich - 2-nd City Hospital, Sofia Katja Stefanova - Institute for Standardization and Sertification, Sofia Erassimir Atanassov - Institute for Microsystems, Sofia Lilija Atanassova - 105 Al. Dimitrov School, Sofia Ljubomir HadJyisKy - Technical University, Sofia Luomila Dimitrova - Institute of Chemical Engineering, Burgas Maria Stefanova-Pavlova - Institute for Microsystems, Sofia Martin Tetev - Faculty of Mathematics, El. Ochridsky Univ. , Sofia HiKo Pehlivanov - Institute for Microsystems, Sofia Favlin Gyurov - Faculty of Mathematics, El. Ochridsky Univ. , Sofia Peter Georgiev - Faculty of Mathematics, El. Ochridsky Univ. , Sofia Flamen Fetkov - HEFTCCHIM Petrochemical Combine, Bourgas Radosvet Todotov - Centre for Scientific Information of Bulg. Acad. of Sciences, Sofia Kossen Petrov - Institute for Microsystems, Sofia Rumen Christov - Institute for Microsystems, Sofia Sergej Bedev - Technical University, Sofia

386

367

Stanislav Pishmanov - Faculty of Mathematics, Kl. Ochridsky Univ. , Sofia Stefan Stefanov - Institute for Microsystems, Sofia Stela Dimitrova - NEFTOCHIM Petrochemical Combine, Bourgas Stojan Mihov - Faculty of Mathematics, Kl. Ochridsky Uhiv. , Sofia Stoian Garbov - Central Laboratory of Automation, Sofia Trajana Kolarova - NEFTOCHIM Petrochemical Combine, Bourgas